Answer :
Answer:
The minimum cost is $17.30c
Step-by-step explanation:
Given
[tex]V = 288[/tex] --- Volume
[tex]C_1 = 0.06[/tex] --- cost of wall paint
[tex]C_2 = 0.16[/tex] ---cost of ceiling paint
Required
Minimum cost of paint
The volume is calculated as:
[tex]V =xyz[/tex]
Substitute 288 for V
[tex]288 =xyz[/tex]
Make z the subject
[tex]z = \frac{288}{xy}[/tex]
The surface area is calculated as:
[tex]Area = 2(yz + xz) + xy[/tex]
Because xy represent the dimension of the ceiling and the opposite of the ceiling (the floor) will not be painted. Hence, it does not require a coefficient of 2
The cost is:
[tex]C = 0.06 * 2(yz + xz) + 0.16 * xy[/tex]
Substitute [tex]z = \frac{288}{xy}[/tex]
[tex]C = 0.06 * 2(y*\frac{288}{xy} + x*\frac{288}{xy}) + 0.16 * xy[/tex]
[tex]C = 0.06 * 2(\frac{288}{x} + \frac{288}{y}) + 0.16 * xy[/tex]
[tex]C = 0.12(\frac{288}{x} + \frac{288}{y}) + 0.16 * xy[/tex]
[tex]C = (\frac{34.56}{x} + \frac{34.56}{y}) + 0.16 * xy[/tex]
[tex]C = (\frac{34.56}{x} + \frac{34.56}{y}) + 0.16 xy[/tex]
Differentiate w.r.t x and y
[tex]C_x = -\frac{34.56}{x^2} + 0.16y[/tex]
[tex]C_y = -\frac{34.56}{y^2} + 0.16x[/tex]
By comparison: [tex]x = y[/tex]
Set them equal to 0
[tex]C_y = -\frac{34.56}{y^2} + 0.16x=0[/tex]
[tex]-\frac{34.56}{y^2} + 0.16x=0\\[/tex]
Substitute x for y
[tex]-\frac{34.56}{x^2} + 0.16x=0[/tex]
[tex]0.16x=\frac{34.56}{x^2}[/tex]
Cross multiply
[tex]0.16x^3 = 34.56[/tex]
[tex]x^3 = \frac{34.56}{0.16}[/tex]
[tex]x^3 = 216[/tex]
Take the cube root of both sides
[tex]x = \sqrt[3]{216}[/tex]
[tex]x = 6[/tex]
[tex]x=y= 6[/tex]
Substitute 6 for x and for y in [tex]C = (\frac{34.56}{x} + \frac{34.56}{y}) + 0.16 xy[/tex]
[tex]C = (\frac{34.56}{6} + \frac{34.56}{6}) + 0.16 * 6* 6[/tex]
[tex]C = (\frac{2*34.56}{6}) + 5.76[/tex]
[tex]C = 11.52 + 5.76[/tex]
[tex]C = 17.28[/tex]
[tex]C = 17.3[/tex] --- approximated