To minimize cost, we need to determine whether it's cheaper to lay the power line underground from A to S and then underwater from S to B, or to lay it underwater directly from A to B.
Let CS = x miles. Then AS = 6 - x miles and SB = 8 + x miles.
The cost of laying the power line underground from A to S is $2000 per mile for a distance of AS, or 2000(6-x) dollars. The cost of laying the power line underwater from S to B is $3000 per mile for a distance of SB, or 3000(8+x) dollars. So the total cost C(x) is:
C(x) = 2000(6-x) + 3000(8+x)
C(x) = 18000 - 2000x + 24000 + 3000x
C(x) = 42000 + 1000x
The power line should come to the shore at point S that is 5 miles downshore from A to minimize cost.
To minimize cost, we need to find the value of x that minimizes C(x). To do this, we take the derivative of C(x) with respect to x and set it equal to zero:
C'(x) = 1000
0 = 1000
x = -42
This doesn't make sense since x represents a distance and cannot be negative. So we know that this is not the minimum.
Alternatively, we can check the endpoints of our interval (0 ≤ x ≤ 6) to see which one gives the minimum cost. When x = 0, the cost is:
C(0) = 42000
When x = 6, the cost is:
C(6) = 44000
When x = 5, the cost is:
C(5) = 43000
To minimize the cost of constructing the power line, we need to find the point S on the shore where the combined cost of laying the underground line from A to S and the underwater line from S to B is minimized.
Let x be the distance from A to S, then the distance from S to B is (6 - x) miles.
Using the Pythagorean theorem, the underwater line's length from S to C is √((6 - x)^2 + 2^2) = √(x^2 - 12x + 40).
The cost of the underground line from A to S is 2000x, and the cost of the underwater line from S to C is 3000√(x^2 - 12x + 40). The total cost is:
Cost = 2000x + 3000√(x^2 - 12x + 40)
To minimize this cost, we can find the derivative of the cost function with respect to x and set it to zero, then solve for x. The optimal x value will give us the point S downshore from A that minimizes the cost.
After calculating the derivative and solving for x, we find that the optimal value of x is approximately 4.24 miles. Therefore, the point S should be approximately 4.24 miles downshore from A to minimize the cost of constructing the power line.
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A line passes through the points (–
3,–
18) and (3,18). Write its equation in slope-intercept form
The equation of the line with given coordinates in slope intercept form is given by y = 6x.
Use the slope-intercept form of the equation of a line,
y = mx + b,
where m is the slope of the line
And b is the y-intercept.
The slope of the line is equals to,
m = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points on the line.
Using the coordinates (-3, -18) and (3, 18), we get,
⇒m = (18 - (-18)) / (3 - (-3))
⇒m = 36 / 6
⇒m = 6
So the slope of the line is 6.
Now we can use the slope-intercept form of the equation of a line .
Substitute in the slope and one of the points, say (-3, -18) to get the y-intercept,
y = mx + b
⇒ -18 = 6(-3) + b
⇒ -18 = -18 + b
⇒ b = 0
So the y-intercept is 0.
Putting it all together, the equation of the line in slope-intercept form is,
y = 6x + 0
⇒ y = 6x
Therefore, the slope intercept form of the line is equal to y = 6x.
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Al has a cylindrical storage container 30 centimeters tall with a diameter of 22 centimeters. How much bird food in cubic centimeters will fit in the container? Use the formula V = Bh and approximate π using 3.14. Round your answer to the nearest tenth.
The amount of bird food in cubic centimeters will fit in the container is
11, 398. 2 cubic centimeters
How to determine the volumeThe formula that is used for calculating the volume of a cylinder is expressed with the equation;
V = π(d/2)²h
Such that the parameters of the given equation are;
V is the volume of the cylinder.d is the diameter of the cylinderh is the height of the cylinderNow, substitute the values into the formula, we have;
Volume = 3.14 (22/2)² 30
divide the values
Volume = 3.14(121)30
Now, multiply the values and expand the bracket
Volume = 11, 398. 2 cubic centimeters
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3⋅50. 2w=720
What is the solution of the equation?
Round your answer, if necessary, to the nearest thousandth
The solution to the equation is w = 285.
How to solve a mathematical equation involving multiplication and variables?To solve the equation 3⋅50 + 2w = 720, we first simplify the left side by multiplying 3 and 50, which gives us 150.
Therefore, the equation becomes 150 + 2w = 720. Next, we isolate the variable term by subtracting 150 from both sides of the equation, resulting in 2w = 570.
To solve for w, we divide both sides of the equation by 2, giving us w = 285.
Therefore, the solution to the equation is w = 285.
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What is the particular solution to the differential equation dy/dx = 2x/y with the initial condition y (5) = 4?
The initial condition y(5) = 4 tells us that we should use the positive square root.
To find the particular solution to the given differential equation, we can use separation of variables. First, we rearrange the equation to get:
y dy = 2x dx
Next, we integrate both sides with respect to their respective variables:
∫y dy = ∫2x dx
This gives us:
y^2/2 = x^2 + C
where C is the constant of integration. To find the value of C, we use the initial condition y(5) = 4:
4^2/2 = 5^2 + C
8 = 25 + C
C = -17
So the particular solution to the differential equation dy/dx = 2x/y with the initial condition y(5) = 4 is:
y^2/2 = x^2 - 17
or
y = ±√(2x^2 - 34)
Note that there are two possible solutions, one with a positive square root and one with a negative square root, but the initial condition y(5) = 4 tells us that we should use the positive square root.
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Today, everything at the store is on sale. the store offers a 20% discount.
a.what percentage will you pay when a store offers a 20% discount?
b.if the regular price of a t-shirt is $18. what is the discount price?
c.if the regular price of a gaming system is $360. what is the sale price?
show what you typed into the calculator:
d.the discount price of a hat is $18. what’s the regular price (price before the coupon)?
a. You will pay 80% of the original price.
b. $14.40 is the discount price of the t-shirt.
c. The sale price of the gaming system is $288.
d. $22.50 is the regular price of the hat.
a. When a store offers a 20% discount, you will pay 80% of the original price. This is because the discount is taken off the original price, leaving you to pay the remaining percentage.
b. If the regular price of a t-shirt is $18, the discount price can be found by multiplying the regular price by the percentage you will pay after the discount, which is 80%.
Discount price = Regular price x (1 - Discount percentage)
Discount price = $18 x (1 - 0.20)
Discount price = $18 x 0.80
Discount price = $14.40
Therefore, $14.40 is the discount price of the t-shirt.
c. If the regular price of a gaming system is $360, the sale price can be found by multiplying the regular price by the percentage you will pay after the discount, which is 80%.
Sale price = Regular price x (1 - Discount percentage)
Sale price = $360 x (1 - 0.20)
Sale price = $360 x 0.80
Sale price = $288
Therefore, the sale price of the gaming system is $288.
d. If the discount price of a hat is $18 and the discount percentage is 20%, we can find the regular price by dividing the discount price by the percentage you will pay after the discount, which is 80%.
Regular price = Discount price / (1 - Discount percentage)
Regular price = $18 / (1 - 0.20)
Regular price = $18 / 0.80
Regular price = $22.50
Therefore, the regular price of the hat is $22.50.
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On the math exam,5 tasks were given. 25% of students solved at least two tasks. Prove that there was at least one task that no more than 12 students solved if 32 students wrote that test
Given that 25% of students solved at least two tasks and there were 32 students who wrote the test, we can prove that there was at least one task that no more than 12 students solved.
There was at least one task that no more than 12 students solved, we can use a proof by contradiction.
Assume that all five tasks were solved by more than 12 students. This means that for each task, there were at least 13 students who solved it. Since there are five tasks in total, this implies that there were at least 5 * 13 = 65 students who solved the tasks.
However, we are given that only 25% of students solved at least two tasks. If we let the number of students who solved at least two tasks be S, then we can write the equation:
S = 0.25 * 32
Simplifying, we find that S = 8.
Now, let's consider the remaining students who did not solve at least two tasks. The maximum number of students who did not solve at least two tasks is 32 - S = 32 - 8 = 24.
If all five tasks were solved by more than 12 students, then the total number of students who solved the tasks would be at least 65. However, the maximum number of students who could have solved the tasks is 8 (those who solved at least two tasks) + 24 (those who did not solve at least two tasks) = 32.
This contradiction shows that our initial assumption is false. Therefore, there must be at least one task that no more than 12 students solved.
Hence, we have proven that there was at least one task that no more than 12 students solved if 32 students wrote the test.
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a) express ∂z/∂u and ∂z/∂v as functions of u & v by using the chain rule and by expressing z directly in terms of u & v before differentiating.
b) evaluate ∂z/∂u and ∂z/∂v at the given (u,v)
z = tan^-1
(x/y) x = ucosv
y= usinv
(u,v) = (1.3, pi/6)
a) To express ∂z/∂u and ∂z/∂v as functions of u and v, we first need to express z directly in terms of u and v. We are given that:
z = tan^-1(x/y)
And that:
x = ucosv
y = usinv
Substituting these expressions for x and y into the equation for z, we get:
z = tan^-1((ucosv)/(usinv))
z = tan^-1(cotv)
Now we can use the chain rule to find ∂z/∂u and ∂z/∂v:
∂z/∂u = ∂z/∂cotv * ∂cotv/∂u
∂z/∂v = ∂z/∂cotv * ∂cotv/∂v
To find ∂cotv/∂u and ∂cotv/∂v, we use the quotient rule:
∂cotv/∂u = -cosv/u^2
∂cotv/∂v = -csc^2v
Substituting these into the chain rule expressions, we get:
∂z/∂u = (-cosv/u^2) * (1/(1+cot^2v))
∂z/∂v = (-csc^2v) * (1/(1+cot^2v))
Simplifying these expressions using trig identities, we get:
∂z/∂u = (-cosv/u^2) * (1/(1+(cosv/usinv)^2))
∂z/∂v = (-1/sinv^2) * (1/(1+(cosv/usinv)^2))
b) To evaluate ∂z/∂u and ∂z/∂v at (u,v) = (1.3, pi/6), we simply plug in these values into the expressions we derived in part (a):
∂z/∂u = (-cos(pi/6)/(1.3)^2) * (1/(1+(cos(pi/6)/(1.3*sin(pi/6)))^2))
∂z/∂v = (-1/sin(pi/6)^2) * (1/(1+(cos(pi/6)/(1.3*sin(pi/6)))^2))
Simplifying these expressions using trig functions, we get:
∂z/∂u = (-sqrt(3)/1.69^2) * (1/(1+(sqrt(3)/1.3)^2))
∂z/∂v = (-4) * (1/(1+(sqrt(3)/1.3)^2))
Plugging in the values and evaluating, we get:
∂z/∂u ≈ -0.5167
∂z/∂v ≈ -1.5045
To answer this question, we'll first express z directly in terms of u and v, and then apply the chain rule to find the partial derivatives ∂z/∂u and ∂z/∂v.
Given:
z = tan^(-1)(x/y)
x = u*cos(v)
y = u*sin(v)
First, let's express z in terms of u and v:
z = tan^(-1)((u*cos(v))/(u*sin(v)))
Now, we can simplify the expression:
z = tan^(-1)(cot(v))
Next, we'll find the partial derivatives using the chain rule:
a) ∂z/∂u:
Since z doesn't have a direct dependence on u, we have:
∂z/∂u = 0
b) ∂z/∂v:
∂z/∂v = -csc^2(v)
Now let's evaluate the partial derivatives at the given point (u,v) = (1.3, π/6):
∂z/∂u(1.3, π/6) = 0
∂z/∂v(1.3, π/6) = -csc^2(π/6) = -4
So, the partial derivatives at the given point are:
∂z/∂u = 0 and ∂z/∂v = -4.
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Gabriella decides to estimate the volume of an orange by modeling it as a sphere. She measures its circumference as 50.2 cm. Find the orange's volume in cubic centimeters. Round your answer to the nearest tenth if necessary.
The answer is 2143.6.
If a cup of coffee has temperature 95 C in a room where theremperature is 20 C, then, according to Newon's Law of Cooling, thetemperature of the coffee after t minutes is T(t) = 20+ 75e-t/50. What is the average temperature of thecoffee during the first half hour?
To find the average temperature of the coffee during the first half hour, we need to find the temperature of the coffee at t = 0 (when the coffee is just brewed) and at t = 30 (after half an hour has passed).
At t = 0, T(0) = 20 + 75e^0/50 = 20 + 75 = 95 C. At t = 30, T(30) = 20 + 75e^-30/50 ≈ 42.5 C.
So, the temperature of the coffee decreases from 95 C to 42.5 C during the first half hour.
The average temperature during this time period can be found by taking the average of the initial and final temperatures:
Average temperature = (95 C + 42.5 C) / 2 = 68.75 C.
Therefore, the average temperature of the coffee during the first half hour is 68.75 C.
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PLEASE WRITE THE EXPRESSION IN FORM OF |x-b|=c
Write the absolute value equations in the form |x-b|=c (where b is a number and c can be either number or an expression) that have the following solution sets:
Please do both problems
All numbers such that x≤−14.
All numbers such that x≥ -1. 3
The vertical distance the Mars Rover Curiosity has traveled is approximately 84.954 meters.
What will be the absolute value equations in the form |x-b|=c for x ≤ -14 and x ≥ -1.3?An absolute value equation is an equation that contains an absolute value expression, which is defined as the distance of a number from zero on the number line. The equation |x-b|=c represents the distance between x and b is c units. To write the absolute value equations in the form |x-b|=c, we need to determine the values of b and c based on the given solution sets.
For the solution set "All numbers such that x ≤ -14", we know that the distance between x and -14 is always a non-negative value. Therefore, the absolute value of (x-(-14)) or (x+14) is equal to the distance between x and -14. Since we want x to be less than or equal to -14, we can set the absolute value expression to be equal to -c, where c is a positive number. Hence, the absolute value equation is |x+14|=-c.
Similarly, for the solution set "All numbers such that x ≥ -1.3", the distance between x and -1.3 is always a non-negative value. Therefore, the absolute value of (x-(-1.3)) or (x+1.3) is equal to the distance between x and -1.3. Since we want x to be greater than or equal to -1.3, we can set the absolute value expression to be equal to c, where c is a positive number. Hence, the absolute value equation is |x+1.3|=c.
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PLS HELP!
Joanna went school supply shopping. She spent $23.25 on notebooks and pencils. Notebooks cost $2.49 each and pencils cost $1.08 each. She bought a total of 15 notebooks and pencils. How many of each did she buy?
Answer: 10 pencils and 5 notebooks.
Step-by-step explanation:
We will create a system of equations using the information given. Let n be equal to the number of notebooks and p be equal to the number of pencils.
She spent $23.25 on notebooks and pencils. Notebooks cost $2.49 each and pencils cost $1.08 each.
$2.49n + $1.08p = $23.25
She bought a total of 15 notebooks and pencils.
n + p = 15
Next, we will solve for p by substituting.
n + p = 15 ➜ n = 15 - p
$2.49n + $1.08p = $23.25
$2.49(15 - p) + $1.08p = $23.25
$37.35 - $2.49p + $1.08p = $23.25
$37.35 - $1.41p = $23.25
-$1.41p = -$14.10
p = 10 pencils
Lastly, we will solve for n by substituting:
n = 15 - p
n = 15 - 10
n = 5
Find the minimum value of the parabola y=x^2+5/2.
Someone please help fast
The minimal value of the given parabola is y = 5/2, which occurs at x = 0.
To discover the minimum value of the given parabola & that's why we need to determine the vertex of the parabola.
The vertex of a parabola in the form of y = ax^2 + bx + c is given by means of (-b/2a, f(-b/2a)).
in the given parabola, a = 1, b = 0, and c = 5/2. consequently, the x-coordinate of the vertex is -b/2a = 0/(2*1) = 0.
To discover the y-coordinate of the vertex & we substitute x = 0 within the given equation:
y = 0^2 + 5/2 = 5/2
Therefore, the minimal value of the given parabola is y = 5/2.
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Given XV is 20 inches, find the length of arc XW. Leave your answer in terms of pi
The arc length XW in terms of pi is (10pi)/3.
To find the length of arc XW, we need to know the measure of the angle XDW in radians.
Since XV is the diameter of the circle, we know that angle XDV is a right angle, and angle VDW is half of angle XDW. We also know that XV is 20 inches, so its radius, XD, is half of that, or 10 inches.
Using trigonometry, we can find the measure of angle VDW:
sin(VDW) = VD/VDW
sin(VDW) = 10/20
sin(VDW) = 1/2
Since sin(30°) = 1/2, we know that angle VDW is 30 degrees (or π/6 radians). Therefore, angle XDW is twice that, or 60 degrees (or π/3 radians).
Now we can use the formula for arc length:
arc length = radius * angle in radians
So the length of arc XW is:
arc XW = 10 * (π/3)
arc XW = (10π)/3
Therefore, the arc length XW in terms of pi is (10π)/3.
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tommy solved the equation x ^²-x-12=0 select the factores of x^-x-12
Find the quotient of
−
18
x
4
y
4
+
36
x
3
y
3
−
24
x
2
y
2
−18x
4
y
4
+36x
3
y
3
−24x
2
y
2
divided by
6
x
y
6xy.
Step-by-step explanation:
To simplify the expression, we can factor out the common factor -6x²y² from each term in the numerator:
-6x²y²(3y² - 6xy + 4x²) / 6xy
We can cancel out the common factor of 6 in both the numerator and denominator:
- x²y(3y² - 6xy + 4x²) / xy
Now we can simplify the expression further by canceling out the common factor of xy in the numerator:
- x(3y² - 6xy + 4x²)
Thus, the quotient of the numerator and denominator is:
- x(3y² - 6xy + 4x²) / 6xy.
A rectangle is inscribed in a circle of radius 5 centimeters. Find the perimeter of the rectangle
The perimeter of the rectangle that is inscribed in a circle of radius 5 cm is 28cm
Let x and y be the side of the rectangle
Diameter of circle = radius × 2
Diameter = 5×2
Diameter = 10
According to the Pythagorean theorem
(Diameter)² = X² + Y²
10² = X² + Y²
X² + Y² = 100
By this equation, possible value of x and y is 6 and 8 respectively only 6 and 8 will satisfy the equation
So, X = 6 and Y = 8
Perimeter = 2(X+Y)
Perimeter = 2(6+8)
Perimeter = 2(14)
Perimeter = 28 cm
perimeter of the rectangle is 28cm
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Find the positive solution, to the nearest tenth, of f(x)
g(x) = -2x + 25.
X≈
Submit
= g(x), where f(x) = 3* - 2 and
The result will be the x value when the difference between the values of f(x) and g(x) is less than 0.1, which will be the positive solution to the nearest tenth of f(x)=g(x).
To find the positive solution, to the nearest tenth, of f(x)=g(x) using Cora's process, the steps are as follows:
Input the initial value of x,if x=0.
Calculate f(x) and g(x):
f(x) = x² - 8 = 0 - 8 = -8
g(x) = 2x - 4 = 0 - 4 = -4
If f(x) is less than g(x), then x should be increased and vice versa.
Increase or decrease x accordingly and calculate the new values of f(x) and g(x).
Keep repeating steps 3 and 4 until the difference between the values of f(x) and g(x) is less than 0.1.
Hence, the result will be the x value when the difference between the values of f(x) and g(x) is less than 0.1, which will be the positive solution to the nearest tenth of f(x)=g(x).
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Cora is using successive approximations to estimate a positive solution to f(x) = g(x), where f(x)=x2 - 8 and g(x)=2x - 4. The table shows her results for different input values of x. Use Cora's process to find the positive solution, to the nearest tenth, of f(x) = g(x)
Please Help!
For Ln=1n∑ni=1i−1n , given Ln as indicated, express their limits as n→[infinity] as definite integrals, identifying the correct intervals
The limit of Ln as n approaches infinity is -1/2, and it can be expressed as the definite integral ∫0¹ (x - 1) dx over the interval [0, 1].
To express the limit of Ln as n approaches infinity as a definite integral, we can use the definition of the definite integral as the limit of a Riemann sum. We can divide the interval [0, 1] into n subintervals of equal width Δx = 1/n, and evaluate Ln as the limit of the Riemann sum:
Ln = 1/n * [f(0) + f(Δx) + f(2Δx) + ... + f((n-1)Δx)]
where f(x) = x - 1 is the function being integrated.
Taking the limit as n approaches infinity, we have:
lim(n→∞) Ln = lim(n→∞) 1/n * [f(0) + f(Δx) + f(2Δx) + ... + f((n-1)Δx)]
= ∫0¹ (x - 1) dx
where we have used the fact that the limit of the Riemann sum is equal to the definite integral of the function being integrated.
Therefore, the limit of Ln as n approaches infinity is equal to the definite integral of (x - 1) over the interval [0, 1].
So,
lim(n→∞) Ln = ∫0¹ (x - 1) dx = [x¹ - x] from 0 to 1
= [1/2 - 1] - [0 - 0]
= -1/2
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Find the following integral results a. So to dz b. C2+ IT x'sir. 'o 1+cos? dx A solid is obtained by rotating the shaded region about the specified line such as the x-axis or the y-axis. Find the volume of the solid
V = ∫2πx f(y) dy volume of the solid
a. The integral of dz is simply z + C, where C is the constant of integration. So the result of integrating dz is:
∫ dz = z + C
b. To find the integral of (C^2 + I∫sin(x))/(1+cos(x)) dx, we can use the substitution u = 1 + cos(x), du/dx = -sin(x), and dx = du/(-sin(x)). Then we have:
∫(C^2 + I∫sin(x))/(1+cos(x)) dx = ∫(C^2 + I∫sin(x))/u (-du/sin(x))
= -I∫(C^2 + I∫sin(x))/u du
= -I(C^2ln|u| + I∫ln|u| sin(x) dx) + C'
= -I(C^2ln|1+cos(x)| - I∫ln|1+cos(x)| sin(x) dx) + C'
where C' is the constant of integration.
c. To find the volume of the solid obtained by rotating the shaded region about the x-axis or the y-axis, we need to use the method of cylindrical shells or disks, respectively.
If we rotate the region about the x-axis, we can use the formula:
V = ∫2πy f(x) dx
where f(x) is the distance from the x-axis to the function y(x) that defines the region. If we have a function y(x) = g(x) - h(x) that defines the region between two curves, then f(x) = g(x) - h(x) and the limits of integration are the x-values where the two curves intersect.
If we rotate the region about the y-axis, we can use the formula:
V = ∫2πx f(y) dy
where f(y) is the distance from the y-axis to the function x(y) that defines the region. If we have a function x(y) = g(y) - h(y) that defines the region between two curves, then f(y) = g(y) - h(y) and the limits of integration are the y-values where the two curves intersect.
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On Saturday mornings, Roger volunteers at the hospital. One Saturday, he answers phone calls at the information desk while the receptionist is away. Then he spends 40 minutes delivering flowers to patients' rooms. In all, Roger volunteers at the hospital for 90 minutes that day. Which equation can you use to find the amount of time t that Roger answers phone calls?
We know that by solving for t, you can find the amount of time Roger spent answering phone calls
You can use the equation t + 40 = 90, where t represents the amount of time Roger answers phone calls at the information desk.
This equation takes into account that Roger spends 40 minutes delivering flowers and the total time he volunteers at the hospital is 90 minutes.
By solving for t, you can find the amount of time Roger spent answering phone calls.
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Question 21
What fraction is equal to 40 % ?
A
B
C
D
45
58
2
1
25
Answer:
2/5
Step-by-step explanation:
To convert a percentage to a fraction, we divide the percentage by 100 and simplify the resulting fraction. For example, to convert 40% to a fraction, we divide 40 by 100 to get 0.4 and then simplify the fraction 2/5. Therefore, 40% is equal to 2/5.
write a paragraph about importance of english language using simple present tense.
Answer:
Step-by-step explanation:
The English language is an art that I am using to convey this message to you. Without this form of communication, we would be unable to talk or write without using another language. We think everyday with this awesome language, and don't think much about the language we think in. English is an amazing language, and I am proud to be able to verbalize it to you today.
Drag each set of dots to the correct location on the dot plot. Each set of dots can be used more than once. Not all sets of dots will be used. Tricia recorded the number of pets owned by each of her classmates. These data points represent the results of her survey. 0, 3, 2, 4, 1, 0, 0, 3, 2, 1, 2, 1, 1, 3, 4, 2, 0, 0, 1, 1, 1, 0, 3 Create a dot plot that represents the data
A dot plot that represent this data set is shown in the image attached below.
What is a dot plot?In Mathematics and Statistics, a dot plot can be defined as a type of line plot that is typically used for the graphical representation of a data set above a number line, especially through the use of crosses or dots.
Based on the information provided about this data points, we can reasonably infer and logically deduce that the number with the highest frequency is 1.
In this scenario, we would use an online graphing calculator to construct a dot plot with respect to a number line that accurately fit the data set.
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Determine if the sequence below is arithmetic or geometric and determine the common difference / ratio in simplest form.
12
,
8
,
4
,
.
.
.
12,8,4,...
This is sequence and the is equal to
Answer: arithmetic. Common difference is -4
Step-by-step explanation:
constantly subtract four to get to the next
You are asked by your teacher to arrange the letters in the word probability regardless of each word 's meaning. in how many ways can you arrange the letter in the word?
[tex]\color{blue}{analysis}[/tex] : the problem involve permutation or combination) of objects
[tex]\color{red}{required}[/tex] : the value that is to be solved in the problem is the____
[tex]\color{pink}{given}[/tex]: the given value is____ which is the_____ of the word probability
[tex]\color{cyan}{formula}[/tex]: we will use the formula______ to soive for the unknown.
solution
The number of ways to arrange the letters in the word "probability" is 11 factorial (11!).
How many ways to arrange?In this problem, we need to arrange the letters in the word "probability." Since the order of the letters matters, we are dealing with permutations of objects.
The value we are trying to solve is the number of ways to arrange the letters. The given value is the word "probability," which has a total of 11 letters. To solve for the unknown, we will use the formula for permutations.
The formula for permutations of objects is n! / (n - r)!, where n is the total number of objects and r is the number of objects being arranged. In this case, we have 11 letters to arrange, so the formula becomes 11! / (11 - 11)!.
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Find the coordinates of the points on the curve ????=1+costheta wherethe tangent line is vertical or horizontalon[0,2????).
To find the coordinates of the points on the curve r = 1 + cos(θ) where the tangent line is vertical or horizontal on the interval [0, 2π), follow these steps:
1. Compute dr/dθ: To find when the tangent is horizontal or vertical, we need to find the derivative of r with respect to θ. Start by differentiating r = 1 + cos(θ) with respect to θ:
dr/dθ = -sin(θ)
2. Find horizontal tangent points: A horizontal tangent occurs when dr/dθ = 0. In this case, -sin(θ) = 0. Solve for θ:
θ = nπ, where n is an integer
Since we're only considering the interval [0, 2π), we have two values of θ: 0 and π. Now, find the corresponding r-values for these points:
r(0) = 1 + cos(0) = 1 + 1 = 2
r(π) = 1 + cos(π) = 1 - 1 = 0
So, the coordinates for horizontal tangents are (2, 0) and (0, π).
3. Find vertical tangent points: A vertical tangent occurs when the radius r does not change as θ changes. Since dr/dθ = -sin(θ), we are looking for values of θ where sin(θ) is undefined. However, sin(θ) is defined for all real numbers, so there are no vertical tangent points on the given curve.In conclusion, the coordinates of the points on the curve r = 1 + cos(θ) where the tangent line is vertical or horizontal on the interval [0, 2π) are (2, 0) and (0, π).
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Assume that sin(x) equals its Maclaurin series for all
X. Use the Maclaurin series for sin (5x^2) to evaluate
the integral
∫ sin (5x)^2 dx
To evaluate the integral ∫sin(5x^2)dx using the Maclaurin series, we first need to find the Maclaurin series for sin(5x^2).
The Maclaurin series for sin(x) is given by:
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
Now, replace x with 5x^2:
sin(5x^2) = (5x^2) - (5x^2)^3/3! + (5x^2)^5/5! - (5x^2)^7/7! + ...
Now we have the Maclaurin series for sin(5x^2). To evaluate the integral ∫sin(5x^2)dx, we integrate term-by-term:
∫sin(5x^2)dx = ∫[(5x^2) - (5x^2)^3/3! + (5x^2)^5/5! - (5x^2)^7/7! + ...]dx
= (5/3)x^3 - (5^3/3!7)x^7 + (5^5/5!11)x^11 - (5^7/7!15)x^15 + ... + C
This is the integral of sin(5x^2) using the Maclaurin series, where C is the constant of integration.
To evaluate the integral ∫ sin (5x)^2 dx, we can use the identity sin^2(x) = (1-cos(2x))/2.
First, we need to find the Maclaurin series for sin (5x^2). Using the formula for the Maclaurin series of sin(x), we have:
sin (5x^2) = ∑ ((-1)^n / (2n+1)!) (5x^2)^(2n+1)
= ∑ ((-1)^n / (2n+1)!) 5^(2n+1) x^(4n+2)
Next, we substitute this series into the integral:
∫ sin (5x)^2 dx = ∫ sin^2 (5x) dx
= ∫ (1-cos(10x)) / 2 dx
= (1/2) ∫ 1 dx - (1/2) ∫ cos(10x) dx
= (1/2) x - (1/20) sin(10x) + C
where C is the constant of integration.
Therefore, using the Maclaurin series for sin (5x^2), the integral of sin (5x)^2 is (1/2) x - (1/20) sin(10x) + C.
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A rectangular fish tank needs to hold 500 gallons, and it needs to be two feet deep. The top will be open. A. Find the width and length of the tank that will use the smallest amount of glass. B. The tank will be filled with enough water so that there will be two inches of head space. Find the weight of the water in the tank
The weight of the water in the tank is approximately 3,809 pounds.
A. To find the width and length of the tank that will use the smallest amount of glass, we need to consider the surface area of the tank. Let's use "x" to represent the length and "y" to represent the width. The formula for the surface area of a rectangular tank is:
Surface Area = 2xy + 2xz + 2yz
Since the top of the tank will be open, we can ignore the surface area of the top. We know that the tank needs to hold 500 gallons and be 2 feet deep, so we can use the formula for the volume of a rectangular tank to solve for one of the variables:
Volume = Length x Width x Depth
500 = xy x 2
xy = 250
Now we can substitute this into the surface area formula and simplify:
Surface Area = 2(250) + 2xz + 2yz
Surface Area = 500 + 2xz + 2yz
To minimize the surface area, we need to differentiate this formula with respect to one of the variables and set it equal to zero. Let's differentiate with respect to x:
d(Surface Area)/dx = 2z
Setting this equal to zero, we get:
2z = 0
z = 0
This doesn't make sense, so let's try differentiating with respect to y:
d(Surface Area)/dy = 2z
Setting this equal to zero, we get:
2z = 0
z = 0
Again, this doesn't make sense. We can conclude that the surface area is minimized when x = y, so the tank should be square. Since xy = 250, we can solve for the side length of the square:
x^2 = 250
x ≈ 15.81 feet
So the tank should be approximately 15.81 feet by 15.81 feet to use the smallest amount of glass.
B. The volume of the water in the tank will be:
Volume = Length x Width x Depth
Volume = 15.81 x 15.81 x 1.67
Volume = 397.25 gallons
Since the tank needs to hold 500 gallons with 2 inches of head space, we can find the weight of the water using the formula:
Weight = Volume x Density
The density of water is approximately 8.34 pounds per gallon, so:
Weight = 397.25 x 8.34
Weight ≈ 3,313.69 pounds
So the weight of the water in the tank will be approximately 3,313.69 pounds.
A. To minimize the amount of glass used for the rectangular fish tank, you'll need to create a tank with equal width and length (a square base). Since the tank needs to hold 500 gallons and is 2 feet deep, you can use the formula: Volume = Length × Width × Depth. Convert 500 gallons to cubic feet (1 gallon ≈ 0.1337 cubic feet), so 500 gallons ≈ 66.85 cubic feet.
66.85 = Length × Width × 2
33.425 = Length × Width
Since the length and width are equal, you can solve for one of the dimensions:
Length = Width = √33.425 ≈ 5.78 feet
So, the tank dimensions will be approximately 5.78 feet by 5.78 feet by 2 feet.
B. To find the weight of the water in the tank, first determine the volume of the water. There will be 2 inches of headspace (2 inches ≈ 0.167 feet), so the water depth is 2 - 0.167 = 1.833 feet. The volume of the water is:
Volume = Length × Width × Depth = 5.78 × 5.78 × 1.833 ≈ 61.05 cubic feet
To find the weight of the water, multiply the volume by the weight of water per cubic foot (62.43 lbs/cubic foot):
Weight = 61.05 × 62.43 ≈ 3,809 lbs
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I tried to do it but it gave me 6.
Answer:
(c) 12.5
Step-by-step explanation:
You want the unknown leg of a right triangle with one leg 10 and hypotenuse 16.
Sanity checkThe triangle inequality tells you the unknown leg of the triangle will have a length between the difference of the two known legs, and the longest leg of the triangle.
Since this is a right triangle, its longest leg is the hypotenuse. The unknown side cannot be longer than that, so must be less than 16.
The difference of the given lengths is ...
16 -10 = 6
so the missing leg must be longer than 6.
Only one answer choice is between 6 and 16: 12.5.
The missing leg length is 12.5 units.
__
Additional comment
If you want to figure the length, you can use the Pythagorean theorem:
c² = a² +b²
16² = 10² +b²
b² = 256 -100 = 156
b = √156 ≈ 12.49 ≈ 12.5
The length of the unknown leg is 12.5 units.
2x + 7 = -1(3 - 2x) solve for X
This linear equation is invalid, the left and right sides are not equal, therefore there is no solution.
What in mathematics is a linear equation?
An algebraic equation with simply a constant and a first-order (linear) component, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to be linear equations. axe+b = 0 is a one-variable example in which a and b are real numbers and x is the variable.
2x + -7 = -1(3 + -2x)
Reorder the terms:
-7 + 2x = -1(3 + -2x)
-7 + 2x = (3 * -1 + -2x * -1)
-7 + 2x = (-3 + 2x)
Add '-2x' to each side of the equation.
-7 + 2x + -2x = -3 + 2x + -2x
Combine like terms: 2x + -2x = 0
-7 + 0 = -3 + 2x + -2x
-7 = -3 + 2x + -2x
Combine like terms: 2x + -2x = 0
-7 = -3 + 0
-7 = -3
Solving
-7 = -3
Couldn't find a variable to solve for.
This equation is invalid, the left and right sides are not equal, therefore there is no solution.
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