How find the values of x when given y?

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

1 Answer

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.