Select the correct answer.
Tracy wants to make a cylindrical model. The height of the cylindrical model, [tex]H(V)[/tex], in inches, is modeled by this equation, where [tex]V[/tex] is the volume of the cylindrical model, in cubic inches.
[tex]H(V)=2.69 \sqrt[3]{V}[/tex]
What is the average rate of change of the model's height when its volume changes from 512 cubic inches to 1,000 cubic inches?
A. The model's height increases by approximately 26.9 inches for every 1 -cubic-inch increase in its volume.
B. The model's height increases by approximately 21.52 inches for every 1 -cubic-inch increase in its volume.
C. The model's height increases by approximately 0.022 inch for every 1-cubic-inch increase in its volume.
D. The model's height increases by approximately 0.011 inch for every 1 -cubic-inch increase in its volume.
Answer (2)
Calculate H ( 512 ) = 2.69 × 3 512 = 21.52 .
Calculate H ( 1000 ) = 2.69 × 3 1000 = 26.9 .
Calculate the average rate of change: 1000 − 512 H ( 1000 ) − H ( 512 ) = 488 26.9 − 21.52 = 0.01102459 .
The model's height increases by approximately 0.011 inch for every 1-cubic-inch increase in its volume.
Explanation
Understanding the Problem We are given the function H ( V ) = 2.69 3 V which models the height of a cylindrical model in inches, where V is the volume in cubic inches. We want to find the average rate of change of the model's height when the volume changes from 512 cubic inches to 1000 cubic inches. The average rate of change is given by the formula: 1000 − 512 H ( 1000 ) − H ( 512 ) First, we need to calculate H ( 512 ) and H ( 1000 ) .
Calculating H(512) and H(1000) Calculate H ( 512 ) :
H ( 512 ) = 2.69 3 512 = 2.69 × 8 = 21.52 Calculate H ( 1000 ) :
H ( 1000 ) = 2.69 3 1000 = 2.69 × 10 = 26.9 Now, we can plug these values into the average rate of change formula.
Calculating the Average Rate of Change The average rate of change is: 1000 − 512 H ( 1000 ) − H ( 512 ) = 1000 − 512 26.9 − 21.52 = 488 5.38 ≈ 0.01102459 Rounding to five decimal places, the average rate of change is approximately 0.01102.
Final Answer The average rate of change of the model's height when its volume changes from 512 cubic inches to 1,000 cubic inches is approximately 0.011 inch per cubic inch. This means that for every 1-cubic-inch increase in volume, the model's height increases by approximately 0.011 inch.
Selecting the Correct Answer Therefore, the correct answer is: The model's height increases by approximately 0.011 inch for every 1 -cubic-Inch increase in its volume.
Examples
Imagine you're designing a series of cylindrical containers for a specific product. The equation H ( V ) = 2.69 3 V helps you determine the height of the container based on its volume. By calculating the average rate of change, you can estimate how much the height will increase for each additional cubic inch of volume. This is useful for optimizing the container's dimensions to minimize material usage while meeting the required volume, ensuring cost-effectiveness and efficient packaging.
The average rate of change of the model's height when its volume changes from 512 to 1,000 cubic inches is approximately 0.011 inches per cubic inch. Therefore, the correct answer is option D. This indicates that for every 1-cubic-inch increase in volume, the height increases very slightly, by about 0.011 inches.
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