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Chapter 4: Problem 47

Growing Tree The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? the tree's cross section area?

Short Answer

Expert verified
About \( 2 / \pi \) inches was the increase in diameter. The increase in the tree's cross section area requires calculations with the given initial diameter and calculated radius.

Step by step solution

01

Calculate the increase in diameter

We start off with the formula for the circumference of a circle, \( C = \pi * d \). In this case, we know that the circumference has increased by 2 inches, so we can set up an equation: \( 2 = \pi * increase in diameter \). Solving for the increase in diameter, we get \( increase in diameter = 2 / \pi \) inches.
02

Calculate the new diameter

Now using the result from step 1, we can calculate the new diameter of the tree which will be equal to the initial diameter 10 inches plus the increase in diameter we calculated, so \( new diameter = 10 + 2 / \pi \) inches.
03

Calculate the new radius

Since the area of the circle is given by \( A = \pi * R^{2} \), we need to find the new radius. The radius is half of the diameter, so \( new radius = (10 + 2 / \pi) / 2 \) inches.
04

Calculate the increase in area

Finally, we calculate the new area of the tree's cross section with the new radius we obtained in step 3 using the formula for the area of a circle \( new area = \pi * (new radius)^{2} \). Subtract the original area \( \pi * (10/2)^{2} \) from the new area to find the increase in area.

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Most popular questions from this chapter

Multiple Choice If \(y=\tan x, x=\pi,\) and \(d x=0.5,\) what does \(d y\) equal? \(\begin{array}{lll}{\text { (A) }-0.25} & {\text { (B) }-0.5} & {\text { (C) } 0} & {\text { (D) } 0.5}\end{array}\) (E) 0.25

Estimating Volume You can estimate the volume of a sphere by measuring its circumference with a tape measure, dividing by 2\(\pi\) to get the radius, then using the radius in the volume formula. Find how sensitive your volume estimate is to a 1\(/ 8\) in. error in the circumference measurement by filling in the table below for spheres of the given sizes. Use differentials when filling in the last column. $$\begin{array}{|c|c|c|}\hline \text { Sphere Type } & {\text { True Radius }} & {\text { Tape Error Radius Error Volume Error }} \\ \hline \text { Orange } & {2 \text { in. }} & {1 / 8 \text { in. }} \\ \hline \text { Melon } & {4 \text { in. }} & {1 / 8 \text { in. }} \\ \hline \text { Beach Ball } & {7 \text { in. }} & {1 / 8 \text { in. }}\end{array}$$

cost, Revenue, and Profit A company can manufacture \(x\) items at a cost of \(c(x)\) dollars, a sales revenue of \(r(x)\) dollars and a profit of \(p(x)=r(x)-c(x)\) dollars (all amounts in thousands). Find \(d c / d t, d r / d t,\) and \(d p / d t\) for the following values of \(x\) and \(d x / d t\) (a) \(r(x)=9 x, \quad c(x)=x^{3}-6 x^{2}+15 x\) and \(d x / d t=0.1\) when \(x=2 .\) (b) \(r(x)=70 x, \quad c(x)=x^{3}-6 x^{2}+45 / x\) and \(d x / d t=0.05\) when \(x=1.5\)

You may use a graphing calculator to solve the following problems. True or False If the radius of a circle is expanding at a constant rate, then its circumference is increasing at a constant rate. Justify your answer.

Production Level Show that if \(r(x)=6 x\) and \(c(x)=\) \(x^{3}-6 x^{2}+15 x\) are your revenue and cost functions, then the best you can do is break even (have revenue equal cost).
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