Question

The length, width and height of a rectangular solid are in the ratio of 3:2:1. If the volume of the solid is 48 cm\(^3\). The total surface area at the solid is :

• A. \(27 \text{ cm}^2\)
• B. \(32 \text{ cm}^2\)
• C. \(44 \text{ cm}^2\)
• D. \(88 \text{ cm}^2\)

Hint:

1. {Dimensions from ratio:} \(l=3x, w=2x, h=x\). 2. {Use volume to find x:} \(V = lwh \implies (3x)(2x)(x) = 6x^3\). Given \(V=48\), so \(6x^3 = 48 \implies x^3 = 8 \implies x=2\). 3. {Actual dimensions:} \(l=6, w=4, h=2\) cm. 4. {Total Surface Area:} \(TSA = 2(lw + wh + hl)\) \(TSA = 2((6)(4) + (4)(2) + (2)(6)) = 2(24+8+12) = 2(44) = 88 \text{ cm}^2\).

Answer 1

The Correct Option is D

Concept: This problem involves a rectangular solid (cuboid) where the dimensions are in a given ratio and the volume is known. We need to find the total surface area. Formulas for a cuboid with length \(l\), width \(w\), and height \(h\):
Volume (\(V\)) = \(l \times w \times h\)
Total Surface Area (\(TSA\)) = \(2(lw + wh + hl)\) Step 1: Express dimensions in terms of a common factor The ratio of length : width : height is 3 : 2 : 1. Let the common factor be \(x\). Then, length \(l = 3x\), width \(w = 2x\), and height \(h = 1x = x\). Step 2: Use the given volume to find \(x\) The volume of the solid is given as \(V = 48 \text{ cm}^3\). Using the formula \(V = lwh\): \[ (3x)(2x)(x) = 48 \] \[ 6x^3 = 48 \] Divide by 6: \[ x^3 = \frac{48}{6} \] \[ x^3 = 8 \] Take the cube root of both sides: \[ x = \sqrt[3]{8} = 2 \text{ cm} \] Step 3: Calculate the actual dimensions Now that we have \(x=2\):
Length \(l = 3x = 3 \times 2 = 6 \text{ cm}\)
Width \(w = 2x = 2 \times 2 = 4 \text{ cm}\)
Height \(h = x = 2 \text{ cm}\) Step 4: Calculate the Total Surface Area (TSA) Using the formula \(TSA = 2(lw + wh + hl)\): \[ TSA = 2((6)(4) + (4)(2) + (2)(6)) \] \[ TSA = 2(24 + 8 + 12) \] \[ TSA = 2(44) \] \[ TSA = 88 \text{ cm}^2 \] The total surface area of the solid is \(88 \text{ cm}^2\).