Question

The total surface area of cylinder having a certain height is the sum of two parts. one of the parts varies directly with radius and the other parts varies directly with the square of the radius. the total surface area is 7200 units when the radius is 30 units and 3600 units when the radius is 20 units. find the total surface area when radius is 10 units?

Answer 1

Given :

Total surface area of cylinder is sum of two parts,

One part varies with radius.

Second part varies with square of radius.

When radius is 30 units then area is 7200 units,

when radius is 20 units then area is 3600 units.

To find :

Area when radius is 10 units.

Solution :

It is given that

The total surface area of cylinder has two parts:

One path is directly proportional to to its radius r,

Let the constant term for this part is a

and

Other part is directly proportional to the square of its radius r².

Let the constant term for this part is b,

so we can write the formula of Total surface area of cylinder as,

Total surface area (TSA)= ar + br²

(eqution 1)

It is given that

1 When radius of cylinder is 30 units then its total surface area is 7200 units

and

2 When radius of cylinder is 20 units then its total surface area is 3600 units.

So, putting the values of first point in equation 1,

we get

$\bf \: 7200 = (a \times 30) + (b \times {30}^{2} ) \\ \bf 7200 = 30a + 900b$

dividing both sides with 30,

a + 30 b = 240

(equation 2)

Putting the values of first point in equation 1,

we get,

$\bf \: 3600 = (a \times 20) + (b \times {20}^{2} ) \\ \bf 3600 = 20a + 400b$

dividing both sides with 20 ,

a + 20 b = 180

(equation 3)

Subtracting equation 2 with 3, we get

(a + 30b) - (a + 20b) = 240 - 180

10b = 60

b = 6,

putting the value of b in equation 3, we get

$\bf \: a + (30 \times 6) = 240 \\ \bf \: a = 240 - 180 = 60$

so,

a = 60,

Putting the values of a and b in equation 1,

TSA = 60r + 6r²

(equation 4)

To get the value of total surface area when radius of cylinder is 10 units,

putting r = 10 in equation 4, we get

$\bf \: TSA = (60 \times 10) + (6 \times {10}^{2} ) \\ \bf \: TSA \: = 600 + 600 = 1200$

So,

Total surface area of cylinder when r = 10 units is = 1200 units.