Question

Consider the following three statements.
Statement-1: The lateral surface area of a cuboid shaped room $=$ the perimeter of the rectangular floor $\times$ height of the room.
Statement-2 : Total surface area of a cylinder $=$ circumference of the circular base $\times$ (diameter of the base $+$ height of the cylinder).
Statement-3 : Curved surface area of a cone $=\frac{1}{2}\times$ slanting height of the cone $x$ circumference of the base of the cone.

Which of the following is correct?

• A. Statements 1 and 3 are true and statement 2 is false.
• B. Statements 1 and 2 are false and statement 3 is true.
• C. Statement 1 and 3 are false and statement 2 is true.
• D. All statements are true.

Answer 1

The correct option is A

Statements 1 and 3 are true and statement 2 is false.

Explanation for correct statement:

Statement $1:$

Step 1: Given data:

We have to prove that,

the lateral surface area of a cuboid shaped room $=$ the perimeter of the rectangular floor $\times$ height of the room

We know that,

Lateral surface area of cuboid$=2(l+b)\times h$

Step 2: Evaluating the given expression:

Let, the length, breadth and height be $l,b,h$ respectively.

$RHS$$=$the perimeter of the rectangular floor $\times$ height of the room

Now, the perimeter of rectangular floor=$2(l+b)$

$RHS$$=2(l+b)h$

$LHS$$=$Lateral surface area of cuboid shaped room$=2(l+b)h$

Thus, $LHS$$=$$RHS$

Statement $1$ is correct.

Statement $3:$

Step 1: Given data

We have to prove that,

Curved surface area of a cone $=\frac{1}{2}\times$ slanting height of the cone $x$ circumference of the base of the cone

We know that,

Curved surface area of cone$=\mathrm{\pi rl}$

Circumference of circle$=2\mathrm{\pi r}$

Step 2: Evaluating the given expression

$LHS$$=$Curved surface area of a cone$=\mathrm{\pi rl}$

Let, $l$ be the slanting height of cone.

We know that, the base of cone is a circle.

So, circumference of the base of the cone $=2\mathrm{\pi r}$

RHS $=\frac{1}{2}\times$ slanting height of the cone $x$ circumference of the base of the cone

RHS$=\frac{1}{2}\times l\times 2\mathrm{\pi r}=\mathrm{\pi rl}=$LHS

Thus, $LHS$$=$$RHS$

Statement $3$ is correct.

Explanation for incorrect statement

Statement 2:

Step 1: Given data:

We have to prove that,

Total surface area of a cylinder $=$ circumference of the circular base $\times$ (diameter of the base $+$ height of the cylinder)

We know that,

Total surface area of cylinder$=2\mathrm{\pi r}(r+h)$

Circumference of circle$=2\mathrm{\pi r}$

Step 2: Evaluating the given expression

$LHS$$=$Total surface area of a cylinder$=2\mathrm{\pi r}(r+h)$, where $r,h$are radius and height of cylinder

Let, $d,h$ be the diameter and height of cylinder respectively.

So, circumference of the base of the cone $=2\mathrm{\pi r}$

RHS $=2\mathrm{\pi r}(\mathrm{d}+\mathrm{h})\ne$ LHS

Thus, $LHS$$\ne$$RHS$

Statement $2$ is incorrect.

Hence, option $\left(A\right)$ is correct.