/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Let \(b\) represent the base dia... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 40

Let \(b\) represent the base diameter of a conifer tree and let \(h\) represent the height of the tree, where \(b\) is measured in centimeters and \(h\) is measured in meters. Assume the height is related to the base diameter by the function \(h=5.67+0.70 b+0.0067 b^{2}\). a. Graph the height function. b. Plot and interpret the meaning of \(\frac{d h}{d b}\).

Short Answer

Expert verified
#Answer# The derivative of the height function, \(\frac{dh}{db}\) = \(0.70 + 0.0134b\), represents the rate at which the height of the tree increases with respect to the increase in its base diameter. It has a positive value, indicating that the height increases as the base diameter increases, and a positive slope, which means that the height increases at a faster rate as the base diameter grows larger.

Step by step solution

01

Writing the height function

We are given the height function as: $$h = 5.67 + 0.70b + 0.0067b^2$$
02

Graphing the height function

To graph the height function, we can use a graphing tool (such as Desmos, GeoGebra, or a graphing calculator). Simply input the function as given and choose a reasonable domain and range (e.g., positive values for \(b\) since it represents the base diameter). After plotting the function, you should see that it is a quadratic function, with the height increasing as the base diameter increases.
03

Calculating the derivative of the height function with respect to base diameter

To find the derivative of the height function with respect to the base diameter \(b\), we can use the power rule for differentiation. The derivative is denoted as \(\frac{dh}{db}\) and is calculated as follows: $$\frac{dh}{db} = \frac{d(5.67+0.70b+0.0067b^2)}{db} = 0 + 0.70 + 0.0067(2)b = 0.70 + 0.0134b$$
04

Plotting the derivative of the height function

Similar to step 2, plot the function \(\frac{dh}{db}=0.70+0.0134b\) using a graphing tool, with an appropriate domain (positive values for \(b\)). You will see that it is a linear function with a positive slope, indicating that as the base diameter increases, the rate of increase in height also increases.
05

Interpreting the meaning of the derivative

The derivative of the height function, \(\frac{dh}{db}\), represents the rate at which the height of the tree increases with respect to the increase in its base diameter. In this case, it has a positive value, indicating that the height increases as the base diameter increases. Moreover, because the slope is positive, it indicates that the height increases at a faster rate as the base diameter grows larger.

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