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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 66 (page 257)

The volume $V$ (in cubic meters) of the hot-air balloon is given by $V (r) = \frac{4}{3} {蟺谤}^{3} .$ If the radius $r$ is function of $t$ as $r (t) = \frac{2}{3} t^{3},$ find the volume $V$ as a function of the time $t$ .

Short Answer

Expert verified

Hence, $V (t) = \frac{32}{81} {蟺迟}^{9}$

Step by step solution

01

Step 1. Given Information

Given that $V (r) = \frac{4}{3} {蟺谤}^{3}$ and $r (t) = \frac{2}{3} t^{3} .$

02

Step 2. Solution

We have to find $V$ as a function of time $t .$

Here, $V (r) = \frac{4}{3} {蟺谤}^{3}$ and $r (t) = \frac{2}{3} t^{3} .$

So, we have to find $V 鈭� r (t) .$

$V 鈭� r (t) = V (r (t)) V 鈭� r (t) = \frac{4}{3} 蟺 {r (t)}^{3} V 鈭� r (t) = \frac{4}{3} 蟺 {\{\frac{2}{3} t^{3}\}}^{3} V 鈭� r (t) = \frac{4}{3} 蟺脳 \frac{8}{27} t^{9} V 鈭� r (t) = \frac{32}{81} {蟺迟}^{9} V (t) = \frac{32}{81} {蟺迟}^{9}$

03

Step 3. Final answer

Hence, $V$ as a function of $t$ is $V (t) = \frac{32}{81} {蟺迟}^{9}$

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the function f is one-to-one. Find its inverse and check your answer.

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