Question

Please help me do this question mathematics expert

Answer 1

Answer:

$\textsf{i)} \quad \text{T} &=2 \pi r^2 + \dfrac{26}{r}$

ii)  radius = 1.27 cm (3 sf)

   height = 2.55 cm (3 sf)

Step-by-step 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:

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

$\implies h=\dfrac{13}{\pi r^2}$

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

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

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

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

Set it to zero, and solve for r:

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

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

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

Therefore,

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