The sum of the altitude and the base radius of a cylinder is #63 cm#. The radius is #4/5# as long as the altitude. Calculate the surface area volume of the cylinder?

Question

The sum of the altitude and the base radius of a cylinder is #63 cm#. The radius is #4/5# as long as the altitude.  Calculate the surface area volume of the cylinder?

Everly Taylor · Accepted Answer

Surface area of cylinder is #5544# sq.cm. and Volume of cylinder is #86240# cc.

Let the altitude be #h#, as radius is #4/5# of height, it is #(4h)/5#. But sum of altitude and radius is #63#, hence #h+(4h)/5=63# or #(5h)/5+(4h)/5=63# or #(9h)/5=63# Therefore #h=63xx5/9=7cancel(63)xx5/cancel9=35# Hence altitude is #35# cm and radius is #63-35=28# cm. Assume pi=22/7#, hence surface area of cylinder given by #pir(r+h)=22/7xx28xx63# = #22/cancel7xx4cancel(28)xx63=88xx63=5544# sq.cm. Volume of cylinder is #pir^2h=22/7xx28xx28xx35# = #22/cancel7xx4cancel(28)xx28xx35=86240# cc.

Mila Anderson · Answer

undefined Let's denote the altitude as $h$ and the base radius as $r$.

Given that the sum of the altitude and the base radius is $63$ cm, we have the equation:
$$h + r = 63$$

Also, the radius is $\frac{4}{5}$ times the altitude, so:
$$r = \frac{4}{5}h$$

Now, we can substitute the expression for $r$ in terms of $h$ into the first equation to solve for $h$:
$$h + \frac{4}{5}h = 63$$
$$1\frac{4}{5}h = 63$$
$$9/5h = 63$$
$$h = \frac{63 \times 5}{9}$$
$$h = 35$$

Now that we have found $h$, we can find $r$:
$$r = \frac{4}{5} \times 35$$
$$r = 28$$

With $h = 35$ and $r = 28$, we can now calculate the surface area and volume of the cylinder.

Surface Area:
$$A = 2\pi rh + 2\pi r^2$$

$$A = 2\pi \times 28 \times 35 + 2\pi \times 28^2$$

$$A = 1960\pi + 1568\pi$$

$$A = 3528\pi$$

Volume:
$$V = \pi r^2h$$

$$V = \pi \times 28^2 \times 35$$

$$V = 27440\pi$$

So, the surface area of the cylinder is $3528\pi$ square cm and the volume is $27440\pi$ cubic cm.