Answer:
[tex]Radius = 1.997\ in[/tex] and [tex]Height = 7.987\ in[/tex]
[tex]Cost = \$1.05[/tex]
Step-by-step explanation:
Given
[tex]Volume = 100in^3[/tex]
[tex]Cost =\$0.014[/tex] -- Top and Bottom
[tex]Cost =\$0.007[/tex] --- Sides
Required
What dimension of the cylinder minimizes the cost
The volume (V) of a cylinder is:
[tex]V = \pi r^2h[/tex]
Substitute 100 for V
[tex]100 = \pi r^2h[/tex]
Make h the subject
[tex]h = \frac{100 }{\pi r^2}[/tex]
The surface area (A) of a cylinder is:
[tex]A = 2\pi r^2 + 2\pi rh[/tex]
Where
[tex]Top\ and\ bottom = 2\pi r^2[/tex]
[tex]Sides = 2\pi rh[/tex]
So, the cost of the surface area is:
[tex]C = 2\pi r^2 * 0.014+ 2\pi rh * 0.007[/tex]
[tex]C = 2\pi r(r * 0.014+ h * 0.007)[/tex]
[tex]C = 2\pi r(0.014r+ 0.007h)[/tex]
Substitute [tex]h = \frac{100 }{\pi r^2}[/tex]
[tex]C = 2\pi r(0.014r+ 0.007*\frac{100 }{\pi r^2})[/tex]
[tex]C = 2\pi r(0.014r+ \frac{0.007*100 }{\pi r^2})[/tex]
[tex]C = 2\pi r(0.014r+ \frac{0.7}{\pi r^2})[/tex]
[tex]C = 2\pi (0.014r^2+ \frac{0.7}{\pi r})[/tex]
Open bracket
[tex]C = 2\pi *0.014r^2+ 2\pi *\frac{0.7}{\pi r}[/tex]
[tex]C = 0.028\pi *r^2+ \frac{2\pi *0.7}{\pi r}[/tex]
[tex]C = 0.028\pi *r^2+ \frac{2 *0.7}{r}[/tex]
[tex]C = 0.028\pi *r^2+ \frac{1.4}{r}[/tex]
[tex]C = 0.028\pi r^2+ \frac{1.4}{r}[/tex]
To minimize, we differentiate C w.r.t r and set the result to 0
[tex]C' = 0.056\pi r - \frac{1.4}{r^2}[/tex]
Set to 0
[tex]0 = 0.056\pi r - \frac{1.4}{r^2}[/tex]
Collect Like Terms
[tex]0.056\pi r = \frac{1.4}{r^2}[/tex]
Cross Multiply
[tex]0.056\pi r *r^2= 1.4[/tex]
[tex]0.056\pi r^3= 1.4[/tex]
Make [tex]r^3[/tex] the subject
[tex]r^3= \frac{1.4}{0.056\pi }[/tex]
[tex]r^3= \frac{1.4}{0.056 * 3.14}[/tex]
[tex]r^3= \frac{1.4}{0.17584}[/tex]
[tex]r^3= 7.96178343949[/tex]
Take cube roots of both sides
[tex]r= \sqrt[3]{7.96178343949}[/tex]
[tex]r= 1.997[/tex]
Recall that:
[tex]h = \frac{100 }{\pi r^2}[/tex]
[tex]h = \frac{100 }{3.14 * 1.997^2}[/tex]
[tex]h = \frac{100 }{12.52}[/tex]
[tex]h = 7.987[/tex]
Hence, the dimensions that minimizes the cost are:
[tex]Radius = 1.997\ in[/tex] and [tex]Height = 7.987\ in[/tex]
To calculate the cost, we have:
[tex]C = 2\pi r(0.014r+ 0.007h)[/tex]
[tex]C = 2* 3.14 * 1.997 * (0.014*1.997+ 0.007*7.987)[/tex]
[tex]Cost = \$1.05[/tex]
