Question

The total surface area of a right circular cylinder is 1683 cm\[^2\] and its height is 15 cm, then the perimeter of the base is (in cm):

Answer 1

Hint: First, we will assume that the radius of the given cylinder is \[r\] and then use the formula to calculate the total surface area of cylinder is \[2\pi r\left( {r + h} \right)\], where \[r\] is the radius of the base and \[h\] is the height of the cylinder to find radius of the base. Then we will formula to calculate the perimeter of the circle is \[2\pi r\], where \[r\] is the radius of the base of the cone.

Complete step-by-step answer:
It is given that the height of the cylinder \[h\]is 15 cm and the total surface area is \[1683{\text{ c}}{{\text{m}}^2}\].
Let us assume that the radius of the given cylinder is \[r\].

We know that the formula to calculate the total surface area of a cylinder is \[2\pi r\left( {r + h} \right)\], where \[r\] is the radius of the base and \[h\] is the height of the cylinder.
From the above formula of the total surface area of the cylinder and the value of total surface area, we get
\[ \Rightarrow 2\pi r\left( {r + h} \right) = 1683\]
Substituting the value of \[\pi \] and \[h\] in the above equation, we get
\[
   \Rightarrow 2 \times \dfrac{{22}}{7} \times r\left( {r + 15} \right) = 1683 \\
   \Rightarrow \dfrac{{44}}{7} \times r\left( {r + 15} \right) = 1683 \\
 \]
Multiplying the above equation by \[\dfrac{7}{{44}}\] on each of the sides, we get
\[
   \Rightarrow r\left( {r + 15} \right) = \dfrac{{1683 \times 7}}{{44}} \\
   \Rightarrow {r^2} + 15r = 267.75 \\
 \]
Subtracting the above equation by \[267.75\] on both sides, we get
\[ \Rightarrow {r^2} + 15r - 267.75 = 0\]
Factoring the above equation, we get
\[
   \Rightarrow {r^2} - 10.5r + 25.5r - 267.75 = 0 \\
   \Rightarrow \left( {r - 10.5} \right)\left( {r + 25.2} \right) = 0 \\
 \]
\[ \Rightarrow r - 10.5 = 0\] or \[r + 25.5 = 0\]
\[ \Rightarrow r = 10.5{\text{ or }} - 25.5\]
Since the radius can never be negative, the negative value of \[r\] is discarded.
Thus, the radius of the base of the cylinder is \[10.5\] cm.

We will use the formula to calculate the perimeter of the circle is \[2\pi r\], where \[r\] is the radius of the base of the cone.

Substituting the values of \[r\] in the above formula, we get
\[
  {\text{Perimeter of base}} = 2 \times \dfrac{{22}}{7} \times 10.5 \\
   = 66{\text{ cm}} \\
 \]
Thus, the perimeter of the base is 66 cm.

Note In solving these types of questions, you should be familiar with the formula of total surface area of cylinder and perimeter of the circle. Some students use the formula of curved surface area instead of the total surface area of the cylinder, which is wrong. Then use the given conditions and values given in the question, and substitute the values in this formula, to find the required value.