Question

How find the values of x when given y?

Can someone please explain to me how to do question 5e? Thank you!

Answer 1

a. V=length * width * height`=(84-2x)*(40-2x)*x`
b. Maximum value of `x=8.537729734`
f. Maximum volume `V=13,098.71 cm^3`

Explanation:

The solution is as follows

if square of size x is to be made at the corners then the dimension of the box is
length `=(84-2x)`
width `=(40-2x)`
height `=x`
volume `=(84-2x)*(40-2x)*x`

(a) volume `V=4x^3-248x^2+3360x`

by the Calculus tool using derivatives

`V'=12x^2-496x+3360x`
set `V'=0`

Solve the quadratic equation
`12x^2-496x+3360x=0`

and `x_1=32.7956036` and
`x_2=8.537729734`

b. Maximum value of `x=8.537729734`

which gives the maximum volume

(c) For the graph
https://www.desmos.com/calculator/rzv6yekowm

(d) `x=2` and `V=4(2)^3 -248(2)^2+3360(2)=5760`
`x=6` and `V=4(6)^3 -248(6)^2+3360(6)=12096`
`x=8` and `V=4(8)^3 -248(8)^2+3360(8)=13056`
`x=10` and `V=4(10)^3 -248(10)^2+3360(10)=12800`

(e) Solve the equation with `V=10000`
`4x^3-248x^2+3360x=10000`
`4x^3-248x^2+3360x-10000=0`

Using scientific graphics calculator the values are

`x_1=44.319`
`x_2=13.503`
`x_3=4.177`

(f) Maximum volume
`V_max=4(8.53772973)^3-248(8.53772973)^2+3360(8.53772973)`

`V_max=13,098.71`

God bless..... I hope the explanation is useful.