Answer:
Step-by-step explanation:
4:1 and 6:1
1000 ml of lemonade contains 250 ml of lemon juice.
How much lemon juice does 600 ml of lemonade contain?
how do i work this out
By finding the ratio of total volume to lemmon juice, we can see that In 600ml of juice we have 150ml of lemmon juice.
How much lemon juice does 600 ml of lemonade contain?We know that 1000 ml of lemonade contains 250 ml of lemon juice. Then the ratio to total volume to lemon juice is:
R = 250/1000 = 0.25
Then in 600ml of juice, we will have 0.25 times that of lemmon juice, it gives:
0.25*600ml = 150 ml
In 600ml of juice we have 150ml of lemmon juice.
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Use calculus to find the area A of the triangle with the given vertices. (0, 6), (2, −3), (3, 4)
The area of the triangle with the given vertices is 11.5 square units.
Define the area of triangle by vertices?The area of a triangle can be calculated using the coordinates of its vertices using the following formula.
To find the area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), we can use the following formula:
A = (x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂))/2
Using this formula with the given vertices (0, 6), (2, −3), and (3, 4), we get:
A = (0(-3 − 4) + 2(4 − 6) + 3(6 − (-3)))/2
A = (0 - 4 + 27)/2
A = 23/2
A = 11.5
Therefore, the area of the triangle with the given vertices is 11.5 square units.
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HEEELLLLPPP!
Whoever answers right will get brainliest!!!!!!!!!
Which relation is a function?
Answer:
Step-by-step explanation:
Pre-SolvingWe are given four graphs, and we want to determine which one is a function.
When given graphs, we can use the vertical line test to determine if a graph is a function or not. If the vertical line hits only one point on the graph, it means that it is a function.
If the vertical line hits more than one point, it means it is not a function.
This is because a function must have one unique output for every input. In other words, if we substitute 3 into a function, it cannot equal both 2 AND 4.
SolvingWe can draw a vertical line down each graph. Feel free to use an online annotating tool to do this, or simply use your finger to draw a vertical line down each graph.
On the first one, we can see that if we draw a vertical line down it, the vertical line hits only one point on the graph. This means that the graph shows a function.
On the second graph, if we draw a vertical line through it, we can see it hits more than one point on the graph. This means the graph is not a function.
On the third graph, if we draw a vertical line through it, we can see it also hits more than one point on the graph, so it is not a function.
The final graph is a vertical line itself. If we draw a vertical line through it, all of the points will be shared; the line hits all of the points in the graph, so it is not a function.
This means that the first graph is a function.
See below for annotations.
Given the following list of values, is the mean, median, or mode likely to be the best measure of the center for the data set?
33, 33, 33, 31, 31, 34, 32, 31
Select the correct answer below:
The mode, mean, and median are relatively close together (mode=33, mean=32.125, median=32), we can conclude that this data set is symmetric and does not have any significant outliers.
What is statistics?The area of mathematics known as statistics is concerned with the gathering, examination, interpretation, presentation, and arrangement of data.
The mean, median, and mode are all measures of central tendency that describe the center of a data set in different ways.
The mean is calculated by dividing the total number of values in a data collection by their sum.
The median is the median of the data set when the values are ordered. It is not affected by outliers.
The value that appears most frequently in a data collection is the mode.
It may not be unique or exist in a data set.
For the given data set, the mode is likely to be the best measure of center as it is the value that occurs most frequently. In this case, the mode is 33, which occurs three times, whereas all other values occur only once or twice. Therefore, the mode is the most representative value for this data set.
Alternatively, we can also calculate the mean and median to see how they compare with the mode:
The mean is (33+33+33+31+31+34+32+31)/8 = 32.125
The median is the middle value when the values are arranged in order, which is 32.
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0 8. A function f(x) is said to have the jump discontinuity at a point x = a if lim a)x+a+ c) x→a+ f(x) = lim x→a+ f(x) = lim x→a¯ lim X-a f(x). b) lim x→a f(x) = lim x→a¯ _ f(x) f(x) = f(a) d) lim f(x) → +00 x→a+
The correct option for the given question is c) lim x→a¯ f(x) = f(a) when the function f(x) is said to have jump discontinuity at a point x=a.
What is jump discontinuity?Jump continuity is a concept in calculus that describes the behaviour of a function at a specific point where the function jumps from one value to another value without any intermediate values. In other words, a function is considered jump continuous at a point if the function approaches a finite limit from both the left and right sides of that point, but the function values on the left and right sides of the point are not equal.
According to the given information:
The correct notation for the left-hand limit as x approaches a from the left side is lim x→a¯, where the horizontal line above the "a" indicates approaching from the left side.
The statement "lim x→a¯ f(x) = f(a)" means that the limit of f(x) as x approaches a from the left side is equal to the value of f(a) at x = a. This indicates that the function f(x) has a jump discontinuity at x = a, where the function jumps from one value to another value at that specific point.
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Each week Sanji gives a prize to one employee based on rolling an 8-sided solid with faces numbered 1 through 8. Select all of the true statements. Group of answer choices p(factor of 24) = 6/8 p(odd number) = 1/2 p(number less than 8) = 1 p(multiple of 3) = 1/4 p(even number) = 1/8 P(the number 9) = 0
The true statements are:
p(factor of 24) = 6/8p(odd number) = 1/2p(multiple of 3) = 1/4P(the number 9) = 0How to get the true statementsp(factor of 24) = 6/8: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Among these, the numbers 1, 2, 3, 4, 6, and 8 are on the 8-sided die. So, there are 6 favorable outcomes out of 8 possible outcomes. This statement is true.
p(odd number) = 1/2: There are 4 odd numbers (1, 3, 5, 7) on the 8-sided die, so the probability of rolling an odd number is 4/8 = 1/2. This statement is true.
p(number less than 8) = 1: All numbers on the 8-sided die are less than or equal to 8, but not all are less than 8. There are 7 numbers less than 8 (1, 2, 3, 4, 5, 6, 7), so the probability is 7/8, not 1. This statement is false.
p(multiple of 3) = 1/4: There are 2 multiples of 3 on the 8-sided die (3 and 6), so the probability of rolling a multiple of 3 is 2/8 = 1/4. This statement is true.
p(even number) = 1/8: There are 4 even numbers (2, 4, 6, 8) on the 8-sided die, so the probability of rolling an even number is 4/8 = 1/2, not 1/8. This statement is false.
P(the number 9) = 0: The 8-sided die only has numbers 1 through 8, so it is impossible to roll a 9. The probability of rolling a 9 is indeed 0. This statement is true.
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A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of
33 ft/s. Its height in feet after t seconds is given by y = 33t - 20t².
A. Find the average velocity for the time period beginning when t=3 and lasting
.01 s:
4
.005 s:
.002 s:
.001 s:
NOTE: For the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator.
Estimate the instanteneous velocity when t=3.
Answer:
The average velocity for a time period is given by the change in position divided by the change in time. For the time period beginning when t=3 and lasting 0.01 seconds, we have:
Initial time: t1 = 3 secondsFinal time: t2 = 3.01 secondsInitial position: y1 = 33t1 - 20t1^2 = 9 feetFinal position: y2 = 33t2 - 20t2^2 = 8.397 feetThe change in position is y2 - y1 = -0.603 feet, and the change in time is t2 - t1 = 0.01 seconds. Therefore, the average velocity for this time period is:
average velocity = change in position / change in time = (-0.603 feet) / (0.01 seconds) ≈ -60.3 feet/second
Similarly, for the time periods beginning when t=3 and lasting 0.005 seconds, 0.002 seconds, and 0.001 seconds, we have:
Time period of 0.005 seconds:Initial time: t1 = 3 secondsFinal time: t2 = 3.005 secondsInitial position: y1 = 33t1 - 20t1^2 = 9 feetFinal position: y2 = 33t2 - 20t2^2 = 8.8025 feetChange in position: y2 - y1 = -0.1975 feetChange in time: t2 - t1 = 0.005 secondsAverage velocity: (-0.1975 feet) / (0.005 seconds) = -39.5 feet/secondTime period of 0.002 seconds:Initial time: t1 = 3 secondsFinal time: t2 = 3.002 secondsInitial position: y1 = 33t1 - 20t1^2 = 9 feetFinal position: y2 = 33t2 - 20t2^2 = 8.8808 feetChange in position: y2 - y1 = -0.1192 feetChange in time: t2 - t1 = 0.002 secondsAverage velocity: (-0.1192 feet) / (0.002 seconds) = -59.6 feet/secondTime period of 0.001 seconds:Initial time: t1 = 3 secondsFinal time: t2 = 3.001 secondsInitial position: y1 = 33t1 - 20t1^2 = 9 feetFinal position: y2 = 33t2 - 20t2^2 = 8.9408 feetChange in position: y2 - y1 = -0.0592 feetChange in time: t2 - t1 = 0.001 secondsAverage velocity: (-0.0592 feet) / (0.001 seconds) = -59.2 feet/secondTo estimate the instantaneous velocity when t=3, we can calculate the derivative of the position function with respect to time:
y = 33t - 20t²
dy/dt = 33 - 40t
At t=3, we have:
dy/dt = 33 - 40(3) = -87
Therefore, the estimated instantaneous velocity when t=3 is -87 feet/second.
Find the volume of the following. I need all the answers of the ones not filled out.
So the volume of the cylindrical pool is approximately 1884.96 cubic feet.
So the volume of the soccer ball is approximately 329.59 cubic inches.
So the volume of the small container is approximately 56.55 cubic inches.
So the volume of the tank is 63 cubic feet.
What is volume?In mathematics, volume is a measure of the amount of space that a three-dimensional object occupies. It is typically expressed in cubic units, such as cubic meters, cubic centimeters, cubic feet, or cubic inches. The volume of a simple shape, such as a cube, rectangular prism, cylinder, or sphere, can be calculated using a specific formula based on its dimensions.
Here,
13. The diameter of the cylindrical pool is 20 feet, which means the radius is 10 feet (half of the diameter). The depth of the pool is 6 feet. The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height or depth. Substituting the given values, we get:
V = π(10 ft)²(6 ft)
≈ 1884.96 cubic feet
14. The soccer ball has a radius of 4.3 inches. The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius. Substituting the given value, we get:
V = (4/3)π(4.3 in)³
≈ 329.59 cubic inches
15. The conical container has a radius of 3 inches and a height of 6 inches. The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height. Substituting the given values, we get:
V = (1/3)π(3 in)²(6 in)
≈ 56.55 cubic inches
16. The tank is in the shape of a pyramid, which means it has a rectangular base with length 3 feet and width 7 feet, and a height of 9 feet. The volume of a pyramid is given by the formula V = (1/3)Bh, where B is the area of the base and h is the height. The base of the tank is a rectangle, so its area is given by the formula A = lw, where l is the length and w is the width. Substituting the given values, we get:
A = (3 ft)(7 ft)
= 21 square feet
V = (1/3)(21 sq ft)(9 ft)
= 63 cubic feet
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Please help!!!!!!!!!
Answer:
a) (2x+9) + (3x-4) + 36 = 180
b) x = 27.8
c) 79.4°
Step-by-step explanation:
a) (2x+9) + (3x-4) + 36 = 180
This is because all 3 angles of a triangle equal to 180 degrees.
b) Firstly add like terms
5x + 41 = 180
Subtract 41 from both sides.
5x = 139
Divide 5 on both sides.
x = 27.8
c) <GTN = 3x-4
Plug in x = 27.8
3(27.8)-4 = 79.4
<GTN is 79.4 degrees
Need help will give brainliest and 5 stars! :)
Answer:
The answer is -3
Step-by-step explanation:
log(1/1000)=x
log 10(1/1000)=x
10^x=(1/1000)
10^x=10⁰/10³
10^x=10^-3
x= -3
Answer:
-3
Step-by-step explanation:
I did the test
Hope this helps :)
(PLS MARK BRAINLIEST)
For the following data set, what is the mode? Data set: A, B, A, B, C, D, E, G, E, A, B
Answer:
A
Step-by-step explanation:
the answer would be a because it is shown more than the others
El radio de la lente circular de una lupa es de 4 centímetros. ¿Cuál es el área, en centímetros cuadrados, del vidrio?
Por lo tanto, el área del vidrio de la lupa es de aproximadamente 50.24 centímetros cuadrados.
What is el área?
"El área" is a Spanish phrase that translates to "the area" in English. "The area" is a mathematical term that refers to the measure of the size of a two-dimensional region, usually measured in square units such as square centimeters (cm²) or square meters (m²). The area of a figure can be calculated using various mathematical formulas, depending on the shape of the figure.
El área del vidrio de una lupa circular se puede calcular usando la fórmula del área del círculo, que es A = πr², donde "A" es el área, "π" es una constante aproximadamente igual a 3.14 y "r" es el radio de la lupa.
En este caso, el radio de la lupa es de 4 centímetros. Sustituyendo este valor en la fórmula, obtenemos:
A = πr²
A = 3.14 x 4²
A = 3.14 x 16
A = 50.24
Por lo tanto, el área del vidrio de la lupa es de aproximadamente 50.24 centímetros cuadrados.
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The equation f(t) = -16t2 + 5t +11 represents the motion of a ball being thrown where t is the time after being thrown. What is the maximum height of the ball? Type only a number (rounded to 1/10th if needed)
Answer:
To find the maximum height of the ball, we need to determine the vertex of the parabolic function f(t) = -16t^2 + 5t + 11. The vertex of a parabola is the point where the function reaches its maximum or minimum value.
The x-coordinate of the vertex is given by the formula x = -b/2a, where a and b are the coefficients of the quadratic function. In this case, a = -16 and b = 5, so the x-coordinate of the vertex is:
t = -b/2a = -5/(2*(-16)) = 0.15625
To find the y-coordinate of the vertex, we can substitute t = 0.15625 into the function f(t):
f(0.15625) = -16(0.15625)^2 + 5(0.15625) + 11 ≈ 11.36
Therefore, the maximum height of the ball is approximately 11.4 (rounded to 1/10th).
1A. Draw a Venn diagram for (A–B) nC
Answer:
Draw three overlapping circles to represent the sets A, B, and C.
Label the regions inside each circle with the name of the corresponding set (i.e., A, B, and C).
Shade the region that represents the set A - B, which includes all the elements that belong to A but not to B.
Finally, shade the region that represents the intersection of (A - B) and C, which includes all the elements that belong to both (A - B) and C.
The resulting diagram should show three overlapping circles, with the region for A shaded but with a part removed (to represent the exclusion of elements in B from A), and the region for (A - B) n C shaded within that.
Obtain the HCF of 420 and 272 by using Euclid's division algorithm and verify the using fundamental therorem of arthmetic
The HCF of 420 and 272 is 4.
To find the HCF of 420 and 272 using Euclid's division algorithm, we can proceed as follows:
Step 1: Divide 420 by 272
420 = 272 x 1 + 148
Step 2: Divide 272 by 148
272 = 148 x 1 + 124
Step 3: Divide 148 by 124
148 = 124 x 1 + 24
Step 4: Divide 124 by 24
124 = 24 x 5 + 4
Step 5: Divide 24 by 4
24 = 4 x 6 + 0
So, we can see that the last non-zero remainder obtained by Euclid's division algorithm is 4. Therefore, the HCF of 420 and 272 is 4.
Now, let's verify this using the Fundamental Theorem of Arithmetic.
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of the factors.
The prime factorization of 420 is:
420 = 2^2 x 3 x 5 x 7
The prime factorization of 272 is:
272 = 2^4 x 17
To find the HCF using the prime factorizations, we need to take the product of the common prime factors with the smallest exponent. In this case, the only common prime factor is 2, and it appears with an exponent of 2 in 420 and an exponent of 4 in 272. So, the HCF is:
HCF(420, 272) = 2^2 = 4
This matches the result obtained using Euclid's division algorithm. Therefore, we have verified that the HCF of 420 and 272 is indeed 4.
pls help 7-4 additional practice
The areas of the figures are 72 square centimeter and 81 square inches
Calculating the area of the figuresFigure 3
From the question, we have the following parameters that can be used in our computation:
The composite figure
The total area of the composite figure is the sum of the individual shapes
So, we have
Area = 9 * (12 - 4 - 4) + 12 * 3
Evaluate
Area = 72
Figure 4
The total area of the composite figure is the difference between the area of the trapezoid and the rectangle
So, we have
Area = 1/2 * (8 + 16) * 8 - 5 * 3
Evaluate
Area = 81
Hence. the area of the figure (3) is 81 square inches
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11 =3+4x. Simplify the answer as much as possible
Answer:
x = 2
Step-by-step explanation:
11 = 3 + 4x
8 = 4x
x = 2
Let's Check
11 = 3 + 4(2)
11 = 3 + 8
11 = 11
So, x = 2 is the correct answer.
What is the area of this figure?
3 mi
3 mi
5 mi
5 mi
4mi
8mi
5 mi
5mi
Answer:
180,000 (OR 90,000 if the figure is a triangle)
Step-by-step explanation:
3 x 3 x 5 x 5 x 4 x 8 x 5 x 5 = 180,000
Area = 180,000
IF THE FIGURE IS A TRIANGLE THEN DIVIDE 180,000 BY 2 AND YOU WILL GET 90,000 AS THE AREA
find the surface area of the prism.
Step-by-step explanation:
There are six sides... 3 pairs of equal sides
2 x ( 4x5 + 4x7 + 5x7) = 166 cm^2
Answer:
166 cm²
Step-by-step explanation:
5x7=35
35x2=70
4x7=28
28x2=56
5x4=20
20x2=40
70+56+40= 166cm²
hope this helps!
Tulips should be planted 3 inches apart to give a full look. You have a trapezoidal plot for a flower garden, as shown in the figure. You plan to put tulips along the parallel sides of the garden. The midsegment to the garden is 10 feet long
Answer: 3 and 14
Step-by-step explanation: x ft = 3ft and x + 4 x = 10 (as you stated) x would be 10 so (10 + 4)ft= 14ft
Jamal is comparing two different kinds of fruit at the store .He wants to buy the one that is the best value per pound. what is the cost per pound for each kind of fruit? Which kind of fruit gives him the best value per pound?
Answer:
apples per pound: 0.25
pears per pound: 0.33
Apples give him the best value per pound
Step-by-step explanation:
to find apple per pound: 4/16= 0.25
to find pears per pound: 7/21= 0.33
What is the particular solution to the differential equation?
Answer:
y = (1/3)(e^(-9(x+2)/(x+1)) -1)
Step-by-step explanation:
You want the particular solution to the differential equation ...
y' = 3(1+3y)/(1+x)^2 with y(-2) = 0
Separation of variablesThe differential equation can be rewritten as ...
[tex]\dfrac{dy}{3(1+3y)}=\dfrac{dx}{(1+x)^2}[/tex]
We can define u = 1+3y, then du = 3 and the left side becomes ...
dy/(3(1+3y)) = (1/9)du/u
and its integral is ...
∫(1/9)du/u = (1/9)ln(u) = (1/9)ln(1 +3y)
The integral of the right side is ...
[tex]\int{(1+x)^{-2}}\,dx=-(1+x)^{-1}+C[/tex]
ResultThen the result of integrating both sides of this rewritten differential equation is ...
[tex]\dfrac{1}{9}\ln{(1+3y)}=-\dfrac{1}{1+x}+C[/tex]
The boundary condition can be used to find C:
[tex]\dfrac{\ln(1+3\cdot0)}{9}=-\dfrac{1}{1-2}+C\\\\0=1+C\\\\C=-1[/tex]
Particular solutionSolving this equation for y, we get ...
[tex]\ln(1+3y)=-9\left(\dfrac{1}{1+x}+1\right)\\\\\\1+3y=e^{\left(-\dfrac{9(x+2)}{x+1}\right)}\\\\\\\boxed{y=\dfrac{e^{\left(-\dfrac{9(x+2)}{x+1}\right)}-1}{3}}[/tex]
What is angle
Enter your answer in the box
Answer:
CAB is 37 degrees
Step-by-step explanation:
90 + 53 = 143
180 - 143 = 37
Based on the line of best fit, which of these statements is true?
A. Each item requires about 12 minutes of production time.
B. The equipment starts producing items after about 25 minutes of
warming up.
C. The equipment starts producing items after about 20 minutes of
warming up.
D. Each item requires about 3 minutes of production time.
The true statement regarding the line of best fit is that C. the equipment starts producing items after about 20 minutes of warming up
What is the like about?The line of best fit may be explained as a straight line which is drawn to pass through a set of plotted data point which gives the best and most approximate relationship between the data points.
The equipment starts producing items after about 25 minutes of warning up. This is false because machine takes 20 minutes to produce the items.
The equipment starts producing items after about 20 minutes of warming up. This is True.
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Prove that heat and light usually go together.
Answer:
Step-by-step explanation:
Infra-red waves commonly known as heat waves and light rays are travel in a speed [tex]3*10^{8} ms^{-1}[/tex] in a vacuum.
Trees in a landscaping company's warehouse are being sold at a monthly rate of 7.3%. This situation can be modeled by an exponential function. The warehouse initially contained 45,000 trees.
Which function can be used to find the number of trees in the warehouse at the end of m months?
An exponential function can be used to calculate the number of trees in the warehouse at the end of any given number of months. This function uses the initial number of trees in the warehouse and the rate of 7.3% to determine the result.
What is function?A function is a block of organized, reusable code that is used to perform a single, related action. Functions provide better modularity for your application and a high degree of code reusing. As you already know, C programming is built around functions. All C programs have at least one function, which is main(). A function is a group of statements that together perform a task. Every C program has at least one function, main(), and all the other functions are called user-defined functions.
The function that can be used to find the number of trees in the warehouse at the end of m months is an exponential function given by:
N(m) = 45,000(1.073)^m
Where N(m) is the number of trees in the warehouse at the end of m months and 45,000 is the initial number of trees in the warehouse. The rate of 7.3% is represented by 1.073, which is the base of the exponential function.
For example, if we want to find the number of trees in the warehouse at the end of 5 months, we can substitute m = 5 into the exponential function and calculate the result. This gives us N(5) = 45,000(1.073)^5 = 60,731. Thus, the warehouse will contain 60,731 trees at the end of 5 months.
In conclusion, an exponential function can be used to calculate the number of trees in the warehouse at the end of any given number of months. This function uses the initial number of trees in the warehouse and the rate of 7.3% to determine the result.
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An exponential function can be used to calculate the number of trees in the warehouse at the end of any given number of months. This function uses the initial number of trees in the warehouse and the rate of 7.3% to determine the result.
What is function?
A function is a block of organized, reusable code that is used to perform a single, related action. Functions provide better modularity for your application and a high degree of code reusing. As you already know, C programming is built around functions. All C programs have at least one function, which is main(). A function is a group of statements that together perform a task. Every C program has at least one function, main(), and all the other functions are called user-defined functions.
The function that can be used to find the number of trees in the warehouse at the end of m months is an exponential function given by:
[tex]N(m) = 45,000(1.073)^m[/tex]
Where N(m) is the number of trees in the warehouse at the end of m months and 45,000 is the initial number of trees in the warehouse. The rate of 7.3% is represented by 1.073, which is the base of the exponential function.
For example, if we want to find the number of trees in the warehouse at the end of 5 months, we can substitute m = 5 into the exponential function and calculate the result. This gives us N(5) = 45,000(1.073)⁵ =60,731.
Thus, the warehouse will contain 60,731 trees at the end of 5 months.
In conclusion, an exponential function can be used to calculate the number of trees in the warehouse at the end of any given number of months. This function uses the initial number of trees in the warehouse and the rate of 7.3% to determine the result.
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Please help!!!!
Find the surface area of the composite figure below to
the nearest tenth.
Answer:
16801.2cm²
Step-by-step explanation:
Cone :
SA = πrl
SA = π · 28 · 45 90-45 = 45
SA = 3958.406744
Cylinder :
SA = 2 · π · r · ( r+h)
SA = 2 · π · 28 ( 28 +45 )
SA = 2 · π · 28 (73)
SA = 12842.83077
12842.83077 + 3958.406744 =
16801.2cm²
Evaluate log_10^3.
a) 100
b) 1, 000
c) 9
d) 3
Answer:
The answer is d.
Step-by-step explanation:
Let m = a number
Let n = a different number
[tex]log(m^n)=nlog(m)[/tex]
[tex]log(10^3)=3*log(10)[/tex]
[tex]log(10)=1[/tex]
[tex]log(10^3)=3[/tex]
solve this problem using leibnitz's theorem with details explanation.
We have shown that dⁿ⁺¹ / d xⁿ⁺¹ (xⁿ ln x) = n!/x using Leibniz's theorem.
What is the Leibniz's theorem?
Leibniz's theorem, also known as the product rule for derivatives, is a rule for finding the nth derivative of a product of two functions. It states that if u(x) and v(x) are functions of x, then the nth derivative of their product
To apply Leibniz's theorem, we need to express the function in terms of a product of two functions: u and v.
Let u = xⁿ and v = ln(x)
Then, du/dx = n*xⁿ⁻¹ and dv/dx = 1/x
Using the formula,
d/dx(xⁿ ln(x)) = u(dv/dx) + v(du/dx)
= xⁿ * (1/x) + ln(x) * n * xⁿ⁻¹
= xⁿ⁻¹ + n*xⁿ⁻¹ ln(x)
To find the nth derivative, we differentiate n times using the product rule.
First, we need to find the first few derivatives:
f(x) = xⁿ⁻¹ + n*xⁿ⁻¹ ln(x)
[tex]f'(x) = nx^{(n-2)} + (n-1)*x^{(n-2)} ln(x)[/tex]
[tex]f''(x) = n(n-2)x^{(n-3)} + 2(n-1)x^{(n-3)} ln(x) - (n-1)x^{(n-3)}/(x^2)[/tex]
[tex]f'''(x) = n(n-2)(n-3)x^{(n-4)} + 3(n-1)(n-2)x^{(n-4)} ln(x) - 3(n-1)x^{(n-4)}/(x^2) - 2(n-1)x^{(n-4)} ln(x)/(x^2)[/tex]
and so on...
The pattern becomes apparent and the nth derivative can be expressed as:
[tex]f^{(n)}(x) = n!/(x^{(n+1)})[/tex]
So,
dⁿ⁺¹ / d xⁿ⁺¹ (xⁿ ln x) = [tex]f^{(n)}(x) = n!/(x^{(n+1)})[/tex]
Hence, we have shown that dⁿ⁺¹ / d xⁿ⁺¹ (xⁿ ln x) = n!/x using Leibniz's theorem.
To learn more about Leibniz's theorem visit:
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I would love some help on this
A ) ⅔ x + 6 = 16
→ ⅔ × 15 + 6 = 16
→ 10 + 6 = 16
→ 16 = 16
Proved
LHS = RHSB ) ⅓ x + 4 = 22
→ ⅓ × 15 + 4 = 22
→ 20 = 2
Not Proved
C) 3x + 15 = 30
→ 3×15+15 = 30
→ 3× 30 =30
→ 90= 30
Not Proved
D) x + 25 = 50
→ 15 + 25 = 50
→ 40 = 50
Not Proved
option a is correct answerSolution -210r -24 = 7r + 12
→ 10r - 7r = 12+24
→ 3r = 36
→ r = 12
Option d is correct answerSolution. -33r -2r-8 = -8r + 10
→ 3r - 2r + 8r = 10+8
→ r + 8r = 18
→ 9r = 18
→ r = 2
Option c is correct answerSolution - 4m/3-18 = m + 6
→ 54- m/ 3 = m + 6
→ 54- m = m + 6×3
→ 54- m = m + 18
Here one side m is positive and one side m is negative so both will cancel .→ 18 - 54
→ -36
