To simplify the expressions and arrange the terms by their degree, we can write:
$\longrightarrow\sf\textbf\:f(x)\:= -x^4\:+\:2x^2\:-\:5x\:+\:3$
$\longrightarrow\sf\textbf\:=\:-x^4 + 2x^2 - x - 4x + 3$
$\longrightarrow\sf\textbf\:=\:-x(x^3 - 2x + 1) - 4x + 3$
$\longrightarrow\sf\textbf\:g(x)\:=\:x^4 + 4x^2 + 2x + 1$
$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x + 1 - 2 + 2$
$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x - 1 + 2$
$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x - 1 + 2$
$\longrightarrow\sf\textbf\:x^4 + 4x^2 + 2x + 1 - 2$
$\longrightarrow\sf\textbf\:(x^2 + 1)^2 - 2$
Therefore, we can express the simplified forms of ${\sf{\textbf{f(x)}}}$ and ${\sf{\textbf{g(x)}}}$ as:
$\longrightarrow\sf\textbf\:f(x)\:=\:-x(x^3 - 2x + 1) - 4x + 3$
$\longrightarrow\sf\textbf\:g(x)\:=\:(x^2 + 1)^2 - 2$
[tex]\huge{\colorbox{black}{\textcolor{lime}{\textsf{\textbf{I\:hope\:this\:helps\:!}}}}}[/tex]
[tex]\begin{align}\colorbox{black}{\textcolor{white}{\underline{\underline{\sf{Please\: mark\: as\: brillinest !}}}}}\end{align}[/tex]
[tex]\textcolor{blue}{\small\textit{If you have any further questions, feel free to ask!}}[/tex]
[tex]{\bigstar{\underline{\boxed{\sf{\textbf{\color{red}{Sumit\:Roy}}}}}}}\\[/tex]
everyone pls answer the questions I posted they are urgent
Answer:
unfortunately there's no questions to be answered
Let a = < 2,3, -1 > and 6 = < - 1,5, k >. - Find k so that a and 6 will be orthogonal (form a 90 degree angle). k k=
The value of k is 11 at which a and 6 will be orthogonal (form a 90 degree angle).
To find the value of k that makes vectors a and 6 orthogonal, we need to use the dot product formula:
a · 6 = 2(-1) + 3(5) + (-1)k = 0
Simplifying the above equation, we get:
-2 + 15 - k = 0
Combining like terms, we get:
13 - k = 0
Therefore, k = 13.
However, we need to check if this value of k makes vectors a and 6 orthogonal.
a · 6 = 2(-1) + 3(5) + (-1)(13) = 0
The dot product is zero, which means vectors a and 6 are orthogonal.
Thus, the final answer is k = 11.
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the relationship between group size and percent woodland appears to be negative and nonlinear. which of the following statements explains such a relationship? responses as the percent of woodland increases, the number of deer observed in a group decreases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group decreases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group increases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group increases at a fairly constant rate. as the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group increases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group increases quickly at first and then more slowly. as the percent of woodland increases, the number of deer observed in a group remains fairly constant.
The statement that explains the negative and nonlinear relationship between group size and percent woodland is given by the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly.
This statement suggests that as the amount of woodland id increases, when the number of deer in a group decreases.
However, the rate of decrease is not constant, but rather decreases more slowly as the percent of woodland increases.
This suggests that there may be some threshold or tipping point.
At which the relationship between group size and percent woodland becomes less pronounced.
This kind of relationship is not uncommon in ecological studies.
Where factors like habitat availability, food availability, and predation risk can all influence animal behavior and population dynamics.
Nonlinear relationships like this one can help researchers better understand complex interplay between these factors and behavior of animals they study.
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Suppose a 4 is rolled on a number cube with sides numbered 1, 2, 3, 4, 5, and 6. The
complement of this event would be rolling a 1, 2, 3, 5, or 6. What is the probability of the
complement, written as a fraction in simplest form?
The probability of rolling any number other than 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is 5/6, which can be written as a fraction in simplest form.
The complement of rolling a 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is rolling any number other than 4, which includes rolling a 1, 2, 3, 5, or 6.
To find the probability of the complement, we need to add up the probabilities of rolling each of these numbers.
Since each number has an equal chance of being rolled, we can find the probability of rolling each number by dividing 1 by the total number of possible outcomes (which is 6, since there are six sides on the cube).
Then, we can add up the probabilities of rolling each of the five numbers in the complement:
P(rolling a 1, 2, 3, 5, or 6) = P(rolling a 1) + P(rolling a 2) + P(rolling a 3) + P(rolling a 5) + P(rolling a 6)
P(rolling any number other than 4) = 1 - P(rolling a 4)
P(rolling any number other than 4) = 1 - 1/6 = 5/6
Therefore, the probability of rolling any number other than 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is 5/6, which can be written as a fraction in simplest form.
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It takes a boat hr to go 12 mi downstream, and 6 hr to return. Find the rate of the boat in still water and the rate of the current
The rate of the boat in still water is 5 miles per hour and rate of the boat in current is 3 miles per hour.
Let us represent the rate of boat in still water hence and rate of boat in current be y. Also, we know that speed = distance/time. Hence, keep the values in formula -
Converting mixed fraction to fraction, time = 3/2 hour
Time = 1.5 hour
1.5 (x + y) = 12 : equation 1
Divide the equation 1 by 3
0.5 (x + y) = 4 : equation 2
6 (x - y) = 12 : equation 3
Divide the equation 3 by 6
(x - y) = 2
x = 2 + y : equation 4
Keep the value of x from equation 4 in equation 2
0.5 (2 + y + y) = 4
1 + y = 4
y = 4 - 1
y = 3 miles/ hour
Keep the value y in equation 4 to get x
x = 2 + 3
x = 5 miles per hour
The rate in still water and current is 5 and 3 miles per hour.
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The complete question is-
It takes a boat 1 (1/2) hr to go 12 mi downstream, and 6 hr to return. Find the rate of the boat in still water and the rate of the current.
Find the derivative of the functions and simplify:
f(x) = (x^3 - 5x)(2x-1)
The derivative of the function f(x) = (x³ - 5x)(2x-1) after simplification is 6x⁴ - 10x³ - 10x².
We apply the product rule and simplify to determine the derivative of,
f(x) = (x³ - 5x)(2x-1).
The product rule is used to determine the derivative of the given function f(x),
h(x) = a.b, then after applying product rule,
h'(x) = (a)(d/dx)(b) + (b)(d/dx)(a).
Applying this for function f,
f'(x) = 6x⁴ - 25x² - 10x³ + 15x²
f'(x) = 6x⁴ - 10x³ - 10x².
Therefore, f'(x) = 6x⁴ - 10x³ - 10x² is the derivative of f(x) after simplifying the function.
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The volume of a rectangle or prism is 12 in. ³ one of the dimensions of the prism is a fraction look at the dimensions of the prism be given to possible answers
The possible dimensions of the rectangular prism having volume = 12 in³, are Length = 2 in, width = 3 in, height = 2/3 in, and Length = 1 in, width = 12 in, height = 1/12 in.
To find the possible dimensions of the prism, we need to consider that the volume of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height of the prism, respectively.
Since the volume of the prism is given as 12 in³, we can write: 12 = lwh
Now, we need to find two sets of dimensions that satisfy this equation, where one of the dimensions is a fraction.
Let's try the first set of dimensions:
l = 2 in
w = 3 in
h = 2/3 in
Plugging these values into the formula for the volume, we get:
V = lwh
V = 2 in × 3 in × 2/3 in
V = 4 in³
This confirms that the volume of the prism is indeed 12 in³, and that one of the dimensions (height) is a fraction.
Now, let's try another set of dimensions:
l = 1 in
w = 12 in
h = 1/12 in
Again, plugging these values into the formula for the volume, we get:
V = lwh
V = 1 in × 12 in × 1/12 in
V = 1 in³
This set of dimensions also satisfies the condition that the volume of the prism is 12 in³, with one of the dimensions (height) being a fraction.
Therefore, the possible dimensions of the prism are:
- Length = 2 in, width = 3 in, height = 2/3 in
- Length = 1 in, width = 12 in, height = 1/12 in.
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A Ferris Wheel at a local carnival has a diameter of 150 ft. And contains 25 cars.
Find the approximate arc length of the arc between each car.
Round to the nearest hundredth. Use π = 3. 14 and the conversion factor:
Use the formula: s = rθ to find the arc length
To find the arc length between each car on the Ferris Wheel, we need to first find the measure of the central angle formed by each car.
The Ferris Wheel has a diameter of 150 ft, which means its radius is half that of 75 ft. We can use the formula s = rθ, where s is the arc length, r is the radius, and θ is the central angle in radians.
Since we have 25 cars on the Ferris Wheel, we can divide the circle into 25 equal parts, each representing the central angle formed by each car.
The total central angle of the circle is 2π radians (or 360 degrees), so each central angle formed by each car is:
(2π radians) / 25 = 0.2513 radians (rounded to four decimal places)
Now we can use this central angle and the radius of the Ferris Wheel to find the arc length between each car:
s = rθ
s = 75 ft * 0.2513
s = 18.8475 ft (rounded to four decimal places)
Therefore, the approximate arc length between each car on the Ferris Wheel is approximately 18.85 ft.
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An electronics retailer offers an optional protection plan for a mobile phone it sells. Customers can choose to buy the protection plan for \$100$100dollar sign, 100, and in case of an accident, the customer pays a \$50$50dollar sign, 50 deductible and the retailer will cover the rest of the cost of that repair. The typical cost to the retailer is \$200$200dollar sign, 200 per repair, and the plan covers a maximum of 333 repairs.
Let X be the number of repairs a randomly chosen customer uses under the protection plan, and let F be the retailer's profit from one of these protection plans. Based on data from all of its customers, here are the probability distributions of X and F:
X=\# \text{ of repairs}X=# of repairsX, equals, \#, start text, space, o, f, space, r, e, p, a, i, r, s, end text 000 111 222 333
F=\text{ retailer profit}F= retailer profitF, equals, start text, space, r, e, t, a, i, l, e, r, space, p, r, o, f, i, t, end text \$100$100dollar sign, 100 -\$50−$50minus, dollar sign, 50 -\$200−$200minus, dollar sign, 200 -\$350−$350minus, dollar sign, 350
Probability 0. 900. 900, point, 90 0. 70. 070, point, 07 0. 20. 020, point, 02 0. 10. 010, point, 01
Find the expected value of the retailer's profit per protection plan sold
Note that the expected value of the retailers profit is - $114. This means he made a loss.
How did we arrive at this ?To find the expected value we must proceed as follows
Expected Value - E(F) is
Probability of F - P(F)
= 100 x ($100 - $200) + (P(F) = $50) x ($50 - $200) + (P(F) = $ -200) x ( $ - 200 - $200) + (P(F) = $- 350) x ($ -350 $ 200)
= (0.9) x (-100) + (0.07 ) x (-150) + (0.01) x (-550) + (0.02) x (-400)
= - 90 - 10.5 - 5.5 -8
E(F) = $ -114
So it is right to state that the expected value of the retailer's profit per protection plan sold is -$114, which is a loss.
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Full Question:
An electronics retailer sells mobile phones with an optional protection plan for $100. In case of an accident, the customer pays a $50 deductible and the retailer covers the rest of the repair cost, which is typically $200 per repair. The protection plan covers a maximum of 333 repairs.
Let X be the number of repairs a randomly chosen customer uses under the protection plan, and let F be the retailer's profit from one of these protection plans. The probability distributions of X and F are:
X = number of repairs: 0 1 2 3
Probability: 0.90 0.07 0.02 0.01
F = retailer profit: $100-$50-$200-$350
Probability: 0.90 0.07 0.02 0.01
The task is to find the expected value of the retailer's profit per protection plan sold.
Penelope invested $89,000 in an account paying an interest rate of 6 1/4% compounded continuously. Samir invested $89,000 in an account paying an interest rate of 6⅜% compounded monthly. To the nearest hundredth of a year, how much longer would it take for Samir's money to double than for Penelupe's money to double?
Answer: -10.57
Step-by-step explanation:
Answer:
0.25 years
Step-by-step explanation:
Penelope invested $89,000 in an account paying an interest rate of 6⅜% compounded continuously.
To calculate the time it would take Penelope's money to double, use the continuous compounding interest formula.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]
As the principal amount is doubled, then A = 2P.
Given interest rate:
r = 6.375% = 0.06375Substitute A = 2P and r = 0.06375 into the continuous compounding interest formula and solve for t:
[tex]\implies 2P=Pe^{0.06375t}[/tex]
[tex]\implies 2=e^{0.06375t}[/tex]
[tex]\implies \ln 2=\ln e^{0.06375t}[/tex]
[tex]\implies \ln 2=0.06375t\ln e[/tex]
[tex]\implies \ln 2=0.06375t(1)[/tex]
[tex]\implies \ln 2=0.06375t[/tex]
[tex]\implies t=\dfrac{\ln 2}{0.06375}[/tex]
[tex]\implies t=10.872896949...[/tex]
Therefore, it will take 10.87 years for Penelope's investment to double.
[tex]\hrulefill[/tex]
Samir invested $89,000 in an account paying an interest rate of 6¹/₄% compounded monthly.
To calculate the time it would take Samir's money to double, use the compound interest formula.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]
As the principal amount is doubled, then A = 2P.
Given values:
A = 2PP = Pr = 6.25% = 0.0625n = 12 (monthly)Substitute the values into the formula and solve for t:
[tex]\implies 2P=P\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]
[tex]\implies 2=\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]
[tex]\implies 2=\left(1+0.005208333...\right)^{12t}[/tex]
[tex]\implies 2=\left(1.005208333...\right)^{12t}[/tex]
[tex]\implies \ln 2=\ln \left(1.005208333...\right)^{12t}[/tex]
[tex]\implies \ln 2=12t \ln \left(1.005208333...\right)[/tex]
[tex]\implies t=\dfrac{\ln 2}{12 \ln \left(1.005208333...\right)}[/tex]
[tex]\implies t=11.1192110...[/tex]
Therefore, it will take 11.12 years for Samir's investment to double.
[tex]\hrulefill[/tex]
To calculate how much longer it would take for Samir's money to double than for Penelope's money to double, subtract the value of t for Penelope from the value of t for Samir:
[tex]\begin{aligned}\implies t_{\sf Samir}-t_{\sf Penelope}&=11.1192110......-10.872896949...\\&= 0.246314066...\\&=0.25\; \sf years\;(nearest\;hundredth)\end{aligned}[/tex]
Therefore, it would take 0.25 years longer for Samir's money to double than for Penelope's money to double.
Find the area under the standard normal distribution curve between z=0 and z=0. 98
The area under the standard normal distribution curve between z = 0 and z = 0.98 is:
0.8365 - 0.5000 = 0.3365
To find the area under the standard normal distribution curve between z = 0 and z = 0.98, we can use a standard normal distribution table or a calculator that can compute normal probabilities.
Using a standard normal distribution table, we can look up the area corresponding to a z-score of 0 and a z-score of 0.98 separately and then subtract the two areas to find the area between them.
The area under the standard normal distribution curve to the left of z = 0 is 0.5000 (by definition). The area under the curve to the left of z = 0.98 is 0.8365 (from the standard normal distribution table).
So the area under the standard normal distribution curve between z=0 and z=0.98 is approximately 0.3365.
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Which expression represents the second partial sum for ? 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1
timed
The expression represents the second partial sum for 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1 is 0.8.
The second partial sum of a sequence refers to the sum of the first two terms of the sequence.
The given sequence is: 2(0.4) + 2(0.4)^2 + 2(0.4)^2 + 2(0.4)^3 + 2(0.4)^0 + 2(0.4)^1
To find the second partial sum, we simply add the first two terms of the sequence:
2(0.4) + 2(0.4)^2 = 0.8
Therefore, the expression that represents the second partial sum for the given sequence 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1 is 0.8.
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(c) Katrina recorded the average rainfall amount, in inches, for two cities over the course of 6 months. City A: {5, 2. 5, 6, 2008. 5, 5, 3} City B: {7, 6, 5. 5, 6. 5, 5, 6} (a) What is the mean monthly rainfall amount for each city? (b) What is the mean absolute deviation (MAD) for each city? Round to the nearest tenth. (c) What is the median for each city?
a) The mean monthly rainfall amount for City A is 334.17 inches and for City B is 5.83 inches.
b) The MAD for City A is 464.28 inches and for City B is 0.46 inches.
c) The median for City A is 5 inches and for City B is 6 inches.
(a) To find the mean monthly rainfall amount for each city, we need to add up all the rainfall amounts and divide by the number of months:
For City A: (5 + 2.5 + 6 + 2008.5 + 5 + 3) / 6 = 334.17 inches
For City B: (7 + 6 + 5.5 + 6.5 + 5 + 6) / 6 = 5.83 inches
(b) To find the mean absolute deviation (MAD) for each city, we need to find the absolute deviations from the mean for each data point, then calculate the average of those absolute deviations:
For City A:
Mean = 334.17 inches
Absolute deviations from the mean: |5 - 334.17| = 329.17, |2.5 - 334.17| = 331.67, |6 - 334.17| = 328.17, |2008.5 - 334.17| = 1674.33, |5 - 334.17| = 328.17, |3 - 334.17| = 331.17
MAD = (329.17 + 331.67 + 328.17 + 1674.33 + 328.17 + 331.17) / 6 = 464.28 inches
For City B:
Mean = 5.83 inches
Absolute deviations from the mean: |7 - 5.83| = 1.17, |6 - 5.83| = 0.17, |5.5 - 5.83| = 0.33, |6.5 - 5.83| = 0.67, |5 - 5.83| = 0.83, |6 - 5.83| = 0.17
MAD = (1.17 + 0.17 + 0.33 + 0.67 + 0.83 + 0.17) / 6 = 0.46 inches
(c) To find the median for each city, we need to arrange the data points in order and find the middle value:
For City A: {2.5, 3, 5, 5, 6, 2008.5}
Median = 5 inches
For City B: {5, 5.5, 6, 6, 6.5, 7}
Median = 6 inches
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‼️WILL MARK BRAINLIEST‼️
The true statements are:
The range for the African-American outline is greater than the range for the Holocaust outline, so there is more variability in the data set for African-American outline.The mean for the Holocaust outline, 38.75, is greater than the mean for the African- American outline, 35, so a student working on the Holocaust project spent more time on the outline on average than a student working on the African-American project.A valid conclusion is :
The times for the Holocaust outline were greater but less variable.What is the range and mean of a data set?The range of a data set is the difference between the highest and lowest values in the set.
To calculate the range, you subtract the smallest value from the largest value.
The mean of a data set, also called the average, is the sum of all the values in the set divided by the total number of values in the set.
To calculate the mean, you add up all the values and divide by the total number of values.
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(1 point) Consider the function f(x, y) = xy + 33 – 48y. f har ? at (-43,0) f has at (0,4). a maximum a minimum a saddle some other critical point no critical point f ha: at (463,0). f has at (0,0). f has ? at (0, –4).
At point (-43,0), f has a maximum. At point (0,4), f has a minimum. At point (463,0), f has a maximum. At point (0,0), f may have a critical point. none of the given points are critical points of the function f (x, y) = xy + 33 - 48y.
Hi! To analyze the critical points of the function f(x, y) = xy + 33 - 48y, we first need to find the partial derivatives with respect to x and y:
fx = ∂f/∂x = y
fy = ∂f/∂y = x - 48
Now, we can analyze the given points:
1. (-43, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this point is not a critical point.
2. (0, 4)
At this point, fx = 4 and fy = 0. Again, both partial derivatives are not equal to 0, so this is not a critical point.
3. (463, 0)
At this point, fx = 0 and fy = 463 - 48 = 415. Since both partial derivatives are not equal to 0, this is not a critical point.
4. (0, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this is not a critical point.
5. (0, -4)
At this point, fx = -4 and fy = -48. Again, both partial derivatives are not equal to 0, so this is not a critical point.
In summary, none of the given points are critical points of the function f(x, y) = xy + 33 - 48y.
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x^2+8x+16 What is the perfect factored square trinomial
Answer:
The perfect factored square trinomial that is equivalent to the expression x^2 + 8x + 16 is:
(x + 4)^2
To see why this is the case, you can expand the expression (x + 4)^2 using the FOIL method:
(x + 4)^2 = (x + 4) * (x + 4)
= x^2 + 4x + 4x + 16
= x^2 + 8x + 16
So, x^2 + 8x + 16 can be factored as (x + 4)^2, which is a perfect square trinomial.
A tank initially contains 200 gal of brine in which 30 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 4 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 3 gal/min. Let y represent the
amount of salt at time t. Complete parts a through e.
At what rate (pounds per minute) does salt enter the tank at time t?
The rate at which salt enters the tank at time t is constant & equal to 8 lb/min,
The rate at which salt enters the tank at time t is equal to the product of the concentration of the incoming brine & the rate at which it enters the tank
At time t, the amount of salt in the tank is y(t), & the volume of the brine in the tank is V(t)-
Therefore, the concentration of salt in the tank at time t is:-
c(t) = y(t) / V(t)
The rate at which brine enters the tank at time t is 4 gal/min, & the concentration of salt in the incoming brine is 2 lb/gal
So the rate at which salt enters the tank at time t is:-
2 lb/gal x 4 gal/min = 8 lb/min
Therefore, the rate at which salt enters the tank at time t is constant & equal to 8 lb/min, regardless of how much salt is already in the tank
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Help me pls this is my last try
2. What is the smallest positive degree angle measure equivalent to tan-¹ (0.724)?
42.2°
31.0°
44.6°
35.9°
Of the money that was paid to a transportation company, 60\%60% went towards wages and 80\%80% of what was left went towards supplies.
If there was \$ 400$400 left after those two expenses, what was the original amount paid?
Amount paid = $
Answer is original amount paid to the transportation company was $800.
Let's work backwards from the final amount of $400 to find the original amount paid to the transportation company.
First, we know that percentage given is 80% of what was left after wages went towards supplies. So, if $400 was left after wages were paid, then:
0.8(400) = $320 went towards supplies.
Next, we know that 60% of the original amount went towards wages. So, if $320 went towards supplies, then the remaining amount that went towards wages was:
0.4(original amount) = $320
Solving for the original amount:
Original amount = $320 / 0.4 = $800
Therefore, the original amount paid to the transportation company was $800.
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Ava has two frogs. This is __
1
3 the number of frogs that Heather
has. How many frogs does Heather have? Draw a diagram to
represent the division. Then write and solve an equation.
The value of n which is the number of frogs Heather has is 6.
What is the number of frogs Heather has?The number of frogs Heather has is calculated as follows;
let the number of frogs Heather has = n
So Ava has 2 fogs, which is equal to 1/3 n.
The value of n which is the number of frogs Heather has is calculated as follows;
(1/3) n = 2
multiply both sides by 3;
n = 3 x 2
n = 6
The division using a diagram, is determined as;
0 0
I I
I I
I I
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Find the maximums and minimums and where they are reached of the function f(x,y)=x2+y2+xy in {(x,y): x^2+y^2 <= 1
(i) Local
(ii) Absolute
(iii) Identify the critical points in the interior of the disk (not the border) if there are any. Say if they are extremes, what kind? Or saddle points, or if we can't know using one method?
To find the maximums and minimums of the function f(x,y)=x^2+y^2+xy in the region {(x,y): x^2+y^2<=1}, we need to use the method of Lagrange multipliers.
First, we need to find the gradient of the function and set it equal to the gradient of the constraint (which is the equation of the circle x^2+y^2=1).
∇f(x,y) = <2x+y, 2y+x>
∇g(x,y) = <2x, 2y>
So, we have the equations:
2x+y = 2λx
2y+x = 2λy
x^2+y^2 = 1
Simplifying the first two equations, we get:
y = (2λ-2)x
x = (2λ-2)y
Substituting these into the equation of the circle, we get:
x^2+y^2 = 1
(2λ-2)^2 x^2 + (2λ-2)^2 y^2 = 1
(2λ-2)^2 (x^2+y^2) = 1
(2λ-2)^2 = 1/(x^2+y^2)
Solving for λ, we get:
λ = 1/2 or λ = 3/2
If λ = 1/2, then we get x = -y and x^2+y^2=1, which gives us the critical points (-1/√2, 1/√2) and (1/√2, -1/√2). We can plug these into the function to find that f(-1/√2, 1/√2) = f(1/√2, -1/√2) = -1/4.
If λ = 3/2, then we get x = 2y and x^2+y^2=1, which gives us the critical point (2/√5, 1/√5). We can plug this into the function to find that f(2/√5, 1/√5) = 3/5.
Therefore, the local maximum is (2/√5, 1/√5) with a value of 3/5, the local minimum is (-1/√2, 1/√2) and (1/√2, -1/√2) with a value of -1/4, and the absolute maximum is also (2/√5, 1/√5) with a value of 3/5, and the absolute minimum is on the border, which occurs at (0,1) and (0,-1) with a value of 0.
There are no critical points in the interior of the disk (not the border) that are not extremes or saddle points.
(i) Local extrema:
To find the local extrema, we first find the partial derivatives of f(x, y) with respect to x and y:
f_x = 2x + y
f_y = 2y + x
Set both partial derivatives equal to zero to find critical points:
2x + y = 0
2y + x = 0
Solving this system of equations, we find that the only critical point is (0, 0).
(ii) Absolute extrema:
To determine whether the critical point is an absolute maximum, minimum, or saddle point, we must examine the second partial derivatives:
f_xx = 2
f_yy = 2
f_xy = f_yx = 1
Compute the discriminant: D = f_xx * f_yy - (f_xy)^2 = 2 * 2 - 1^2 = 3
Since D > 0 and f_xx > 0, the point (0, 0) is an absolute minimum of the function.
(iii) Critical points and their classification:
The only critical point in the interior of the disk is (0, 0). As determined earlier, this point is an absolute minimum. No saddle points or other extrema are present within the interior of the disk.
To find any extrema on the boundary of the disk (x^2 + y^2 = 1), we use the method of Lagrange multipliers. However, as the boundary is not part of the domain specified in the question, we will not delve into that here.
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LaShawn designs websites for local businesses. He charges $25 an hour to build a website, and charges $15 an hour to update websites once he builds them. He wants to earn at least $100 every week, but he does not want to work more than 6 hours each week. What is a possible weekly number of hours LaShawn can spend building websites x and updating websites y that will allow him to obtain his goals?
Answer:
1 hour to build a website
5 hours to update websites
Step-by-step explanation:
x is the hours to build a website
y is the hours to update websites
x + y = 6 ------> 25x + 25y = 150
25x + 15y = 100
10y = 50
y = 5
25x + 15 (5) = 100
25x + 75 = 100
25x = 25
x = 1
So, LaShawn needs 1 hour to build a website and 5 hours to update websites to allow him to reach his goals.
Weights of erasers produced by a certain factory are known to follow the uniform distribution between 31. 5 g and 32. 3 g.
(a) (10 points) erasers produced by this factory are sold in packs of 45. A retailer randomly bought 200 packs. Find the probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31. 95 g.
(b) (10 points) each day, a quality control unit examines the erasers produced by this factory. The unit randomly chooses an eraser from the outputs of this factory and weighs it. This process is repeated 50 times. The unit then records the total number of erasers that were found to weigh at least 31. 7 g. (erasers with weights at least 31. 7 g are called "good" erasers)suppose this unit works for 42 consecutive days. Find the probability that, on average, it finds at least 37. 2 "good" erasers per day
a) The probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31.95 g is approximately 0.0384.
b) The probability that, on average, the unit finds at least 37.2 "good" erasers per day is approximately 0.3133.
a) To solve this problem, we need to use the central limit theorem. According to this theorem, the distribution of sample means becomes approximately normal, regardless of the shape of the population distribution, when the sample size is sufficiently large (usually, n >= 30). In this case, since the sample size is 45, we can assume that the distribution of sample means will be approximately normal.
Now, we need to find the probability that the average weight of at least 15 packs is at least 31.95 g. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:
z = (31.95 - 31.9) / (0.163 / √(45)) = 1.77
Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 1.77. This probability is approximately 0.0384.
b) To solve this problem, we need to use the normal approximation to the binomial distribution. Since each eraser is either "good" or "bad", the number of "good" erasers that the unit finds each day follows a binomial distribution with parameters n = 50 and p = probability of finding a "good" eraser = (32.3 - 31.7)/(32.3 - 31.5) = 0.5.
Now, we need to find the probability that, on average, the unit finds at least 37.2 "good" erasers per day. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:
z = (37.2 - 25) / 25 = 0.488
Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 0.488. This probability is approximately 0.3133.
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What is the sum of the series?
6
X (2k – 10)
k3
The sum of the series under the interval (3, 6) will be negative 4.
Given that:
Series, ∑ (2k - 10)
A series is a sum of sequence terms. That is, it is a list of numbers with adding operations between them.
The sum of the series under the interval (3, 6) is calculated as,
∑₃⁶ (2k - 10) = (2 x 3 - 10) + (2 x 4 - 10) + (2 x 5 - 10) + (2 x 6 - 10)
∑₃⁶ (2k - 10) = - 4 - 2 + 0 + 2
∑₃⁶ (2k - 10) = -4
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If the sides of a rectangle are in the ratio 3:4 and the length of the diagonal is 10 cm, find the length of the sides
Answer:
if the diagonal is 10 then the sides are 3*2 and 4*2 which is 6 and 8 respectively because the diagonal makes it a right angled triangle whereby the the 3,4,5 line steps in, so if the diagonal(hypotenuse) is 10 the 10/5 is 2 then you multiply both 3 and 4 by 2 and that gives you the length of two sides
what is the probability that a random point on AK will be on BE
The probability of the event BE falling on a random point AK is 4/11
What is the probability of an event?A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.
In this problem, we need to determine our sample space;
The sample space = 11
The number of favorable outcomes = 4
The probability of a random point on AK to be on BE will be;
P = 4 / 11
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Solve the equation. ㏒₃(1/9)=2x-1
Enter your answer in the box. Enter a fractional answer as a simplified fraction.
The solution to the given equation which is log₃(1/9) = 2x - 1 is equal to x = -1/2.
To solve the equation log₃(1/9) = 2x - 1, we need to isolate the variable x on one side of the equation. We can start by using the logarithm property that states that the logarithm of a number to a base is equal to the exponent to which the base must be raised to obtain that number. In other words, log₃(1/9) = x if and only if [tex]3^x[/tex] = 1/9.
So, let's rewrite the given equation using this property as follows:
[tex]3^{(log(1/9))[/tex] = [tex]3^{2x-1[/tex]
Simplifying the left-hand side using the logarithm property, we get:
1/9 = [tex]3^{(2x - 1)[/tex]
Now, we can solve for x by taking the logarithm of both sides to base 3:
log₃(1/9) = log₃([tex]3^{(2x - 1)[/tex])
-2 = (2x - 1) * log₃(3)
-2 = 2x - 1
2x = -1
x = -1/2
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The cost C (in dollars) for the care and maintenance of a horse and carriage is C=15x+2000, where x is the number of rides. Write an equation for the revenue R in terms of the number of rides.
The equation for revenue R in terms of the number of rides x is given by R = px, where p is the amount charged per ride (in dollars).
The equation for the revenue R in terms of the number of rides can be derived by multiplying the number of rides with the amount charged per ride.
Let the amount charged per ride be p (in dollars).
Then, the equation for revenue R can be written as R = px.
Note that the amount charged per ride is not given in the problem. It can be assumed that the amount charged is a fixed amount for all the rides.
However, the equation for revenue can still be written in terms of the variable p as R = px.
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Jose created a ball pit for his little sister to play in. he put 40 red balls, 55 purple balls, 45 yellow balls, and 60 green balls into the ball pit. while his sister is playing, one ball rolls out of the pit. what is the probability that the ball is red? 0.17 0.17 0.20 0.20 0.25 0.25 0.40
If he put 40 red balls, 55 purple balls, 45 yellow balls, and 60 green balls into the ball pit. while his sister is playing, one ball rolls out of the pit. Therefore, the probability that the ball that rolled out of the pit is red is 0.2.
The probability of selecting a red ball from the ball pit can be found by dividing the number of red balls by the total number of balls in the pit.
Total number of balls = 40 + 55 + 45 + 60 = 200
Probability of selecting a red ball = Number of red balls / Total number of balls
Probability of selecting a red ball = 40 / 200
Probability of selecting a red ball = 0.2
Therefore, the probability that the ball that rolled out of the pit is red is 0.2.
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Two liters of the Gatorade cost $3.98. How much do 8 liters cost?
Answer:
$15.92
Step-by-step explanation:
We Know
2 liters of Gatorade cost $3.98
How much do 8 liters cost?
We take
3.98 x 4 = $15.92
So, 8 liters cost $15.92
