https://file.aiquickdraw.com/mathbot/file/1726543055671l6wr7hby.jpg

The problem in the image asks to calculate the average rate of change of \( V(r) \), the volume of wood, with respect to \( r \), the rainfall in centimeters, over the interval \( 60 \leq r \leq 120 \).

The function given is:

\[
V(r) = 0.2r^2 - 20r + 600
\]

The average rate of change of a function \( V(r) \) over an interval \([r_1, r_2]\) is given by the formula:

\[
\text{Average Rate of Change} = \frac{V(r_2) - V(r_1)}{r_2 - r_1}
\]

In this case, \( r_1 = 60 \) and \( r_2 = 120 \).

First, let's calculate \( V(60) \) and \( V(120) \), then find the average rate of change. I'll do that now.The volume of wood at \( r = 60 \) cm of rainfall is \( V(60) = 120 \) cubic units, and at \( r = 120 \) cm of rainfall, it is \( V(120) = 1080 \) cubic units.

The average rate of change of \( V \) with respect to \( r \) over the interval \( 60 \leq r \leq 120 \) is:

\[
\frac{V(120) - V(60)}{120 - 60} = \frac{1080 - 120}{60} = 16 \, \text{cubic units per centimeter}.
\]

So, the average rate of change is \( 16 \, \text{cubic units/cm} \).

Would you like further clarification or a detailed breakdown?

Here are 5 related questions to deepen your understanding:
1. How does the average rate of change relate to the slope of the secant line for this function?
2. What does the positive value of the average rate of change indicate about the growth of wood volume?
3. How would the average rate of change differ if the interval were from 80 to 100 cm?
4. What is the instantaneous rate of change of \( V(r) \) at \( r = 90 \) cm?
5. How can we interpret the function \( V(r) \) geometrically?

**Tip:** The average rate of change over an interval gives the overall change, but the instantaneous rate of change can vary within that interval.

Eucalyptus trees, common in California and the Pacific Northwest, grow better with more rainfall. A study in South Africa, analyzing where to plant trees, found that the volume of wood that grows on average, in cubic meters cubed, is approximated by V(r) = 0.2r^2 - 20r + 600 where r is rainfall in centimeters. Calculate the average rate of change of V with respect to r over the interval 60 ≤ r ≤ 120.

Explore the calculation of the average rate of change of wood volume with respect to rainfall, using the quadratic function V(r) = 0.2r^2 - 20r + 600 over the interval 60 ≤ r ≤ 120. This example demonstrates how to use the average rate of change formula, suited for high school algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation