Resolved: "The radius r of a sphere is increasing..."

High School
Algebra
Pre-Algebra
Ratios, rates, proportions

Shirley Thompson

Answered question

2021-12-10

The radius r of a sphere is increasing at a rate of 6 inches per minute.
(a) Find the rate of change of the volume when

r = 11

inches.
(b) Find the rate of change of the volume when

r = 34

Answer & Explanation

usumbiix

Beginner2021-12-11Added 33 answers

Step1
(a)
Given:

\frac{d r}{d t} = 6

inches

r = 11 \in c h e s

Step2
Find the rate of change of volume:
V olume of the sphere

= \frac{4}{3} π r^{3}

\frac{d V}{d t} = \frac{4}{3} (3) π r^{2} \frac{d r}{d t}

= 4 π r^{2} \frac{d r}{d t}

Substitue

r = 11

inches and

\frac{d r}{d t} = 6

inches in above equation

\frac{d V}{d t} = 4 π {(11)}^{2} (6)

= 2904 π \in \frac{.^{3}}{min}

Beverly Smith

Beginner2021-12-12Added 42 answers

Step 3
(b)
Given:
$\frac{d r}{d t} = 6$ inches
$r = 34 i n c h e s$
Find the rate of change of volume:
Substiture $r = 34$ inches and $\frac{d r}{d t} = 6$ inches in $\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$
$\frac{d V}{d t} = 4 π {(34)}^{2} (6)$
$= 27744 π \frac{i n^{3}}{min}$

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