Mathematics, 21.12.2020 05:10 texas101st78

The volume of a sphere is decreasing at a constant rate of 116 cubic centimeters per second. At the instant when the volume of the sphere is 77 cubic centimeters, what is the rate of change of the radius? The volume of a sphere can be found with the equation V=\frac{4}{3}\pi r^3.V= 3
4

πr
3
. Round your answer to three decimal places.

Answers: 1

Show answers

Answers

Answer from: AQUI

$\frac{dr}{dt} = -1.325 \ cm/s$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Equality Properties

Calculus

Derivatives

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Taking Derivatives with respect to time

Step-by-step explanation:

Step 1: Define

Given:

$V = \frac{4}{3} \pi r^3$

$\frac{dV}{dt} = -116 \ cm^3/s$

$V = 77 \ cm^3$

Step 2: Solve for r

Substitute: $77 = \frac{4}{3} \pi r^3$ Isolate r term: $\frac{77}{\frac{4}{3} \pi} = r^3$ Isolate r: $\sqrt[3]{\frac{77}{\frac{4}{3} \pi}} = r$ Evaluate: 2.63917 = r

Rewrite: $r = 2.63917 \ cm$

Step 3: Differentiate

Differentiate the Volume Formula with respect to time t.

Define: $V = \frac{4}{3} \pi r^3$ Differentiate [Basic Power Rule]: $\frac{dV}{dt} = \frac{4}{3} \pi \cdot 3 \cdot r^{3-1} \cdot \frac{dr}{dt}$ Simplify: $\frac{dV}{dt} = 4 \pi r^2 \cdot \frac{dr}{dt}$

Step 4: Find radius rate

Substitute in variables: $-116 \ cm^3/sec = 4 \pi (2.63917 \ cm)^2 \cdot \frac{dr}{dt}$ Isolate dr/dt rate: $\frac{-116 \ cm^3/s}{4 \pi (2.63917 \ cm)^2} = \frac{dr}{dt}$ Evaluate: $-1.3253 \ cm/s = \frac{dr}{dt}$ Rewrite: $\frac{dr}{dt} = -1.3253 \ cm/s$ Round: $\frac{dr}{dt} = -1.325 \ cm/s$

Our radius is decreasing at a rate of -1.325 cm per second.

Answer from: Quest

Add a picture

Answer from: Quest

this linear equation would have just one solution.

step-by-step explanation:

the reason why this would have one solution is that it is an expression with just one variable. since that variable is not raised to a power, it will always have just one solution.

Another question on Mathematics

Mathematics, 21.06.2019 21:40

The sides of a parallelogram are 24cm and 16cm. the distance between the 24cm sides is 8 cm. find the distance between the 16cm sides

Answers: 3

Answer

Mathematics, 21.06.2019 21:50

Free points also plz look my profile and answer really stuff

Answers: 2

Answer

Mathematics, 21.06.2019 23:40

Which point-slope form equation represents a line that passes through (3,-2) with a slope of -4/5

Answers: 1

Answer

Mathematics, 22.06.2019 02:00

Taylor graphs the system below on her graphing calculator and decides that f(x)=g(x) at x=0, x=1, and x=3. provide taylor with some feedback that explains which part of her answer is incorrect and why it is incorrect. f(x)= 2x+1 g(x)= 2x^2+1

Answers: 2

Answer

You know the right answer?

The volume of a sphere is decreasing at a constant rate of 116 cubic centimeters per second. At the...

Questions