Respuesta :
Answer:
[tex]V(m)=300m+125[/tex]
Step-by-step explanation:
Given : A cube has an edge of 5 feet. The edge is increasing at the rate of 4 feet per minute.
To Find: Express the volume of the cube as a function of mm, the number of minutes elapsed.
Solution:
We are given that The edge is increasing at the rate of 4 feet per minute.
let x denotes the length of edge
So, [tex]\frac{dx}{dt}=4[/tex]
Volume of cube = [tex]x^3[/tex]
Where x is the length of side
Differentiating with respect to time
[tex]\frac{dV}{dt}=3x^2 \frac{dx}{dt}[/tex] --A
Now we are given that A cube has an edge of 5 feet.
So, x= 5
And we are also given that [tex]\frac{dx}{dt}=4[/tex]
Substitute the values in A
[tex]\frac{dV}{dt}=3(5)^2 \times4[/tex]
[tex]\frac{dV}{dt}=300[/tex]
Now integrating both sides to express the volume of the cube as a function of mm
[tex]V(m)=300m+c[/tex]
When m = no. of minutes = 0
So, c = initial volume
Initial volume of cube = [tex]5^3 =125 ft^3[/tex]
So, the volume of the cube as a function of mm, the number of minutes elapsed is [tex]V(m)=300m+125[/tex]
The function that gives the volume of the cube after m minutes is given by:
[tex]V(m) = (5 + 4m)^3[/tex]
The volume of a cube of edge e is given by:
[tex]V = e^3[/tex]
In this problem, the edge is represented by a linear function, which starts at 5 feet, and increases by 4 feet each minute, thus:
[tex]e(m) = 5 + 4m[/tex]
Then, the volume is given by:
[tex]V = e^3[/tex]
[tex]V(m) = (5 + 4m)^3[/tex]
A similar problem is given at https://brainly.com/question/23514867
