Question

Please help me do this question mathematics expert

Answer 1

`\textsf{i)} \quad \text{T} &=2 \pi r^2 + (26)/(r)`

ii) radius = 1.27 cm (3 sf)

height = 2.55 cm (3 sf)

Explanation:

`\text{Volume of a cylinder}=\pi r^2 h \quad \text{(where r is radius and h is height)}`

Given volume = 13 cm³, rewrite the equation making h the subject:

`\implies 13=\pi r^2 h`

`\implies h=(13)/(\pi r^2)`

Substitute found expression for h into the Surface Area equation to find the expression for the total surface area, T:

`\begin{aligned}\text{Surface Area of a cylinder} & =2 \pi r^2 + 2 \pi r h\\\implies \text{Surface Area} & =2 \pi r^2 + 2 \pi r \left((13)/(\pi r^2)\right)\\& =2 \pi r^2 + (26 \pi r)/(\pi r^2)\\ \implies \text{T} &=2 \pi r^2 + (26)/(r)\end{aligned}`

To find the radius of the minimum surface area, differentiate T with respect to r:

`\begin{aligned}\text{T}& =2 \pi r^2 +26r^(-1)\\\implies (dT)/(dr) & =(2)2 \pi r^((2-1))+(-1)26r^((-1-1))\\ & =4 \pi r-26r^(-2)\\ & =4 \pi r-(26)/(r^2)\\ & = (4 \pi r^3-26)/(r^2) \end{aligned}`

Set it to zero, and solve for r:

`\begin{aligned}(dT)/(dr) & = 0 \\\implies (4 \pi r^3-26)/(r^2) & = 0 \\\impliles 4 \pi r^3-26 & = 0 \\4 \pi r^3 & = 26 \\r^3 & = (13)/(2 \pi) \\r & = \sqrt[3]{(13)/(2 \pi)} \end{aligned}`

To find the height, substitute the found value of r into the equation for height (found previously):

`\begin{aligned}h & =(13)/(\pi r^2) \\\implies h & =\frac{13}{\pi \left(\sqrt[3]{(13)/(2 \pi)}\right)^2} \\& =2.548499134\end{aligned}`

Therefore,

• radius = 1.274249567... = 1.27 cm (3 sf)
• height = 2.548499134... = 2.55 cm (3 sf)