Introduction / Context:
We are given the ratio of the edges of a cuboid and its total surface area (TSA). Using the TSA formula and ratio scaling, we can determine the exact dimensions. This tests ratio scaling with geometric surface area.
Given Data / Assumptions:
- Edge ratio: l : b : h = 6 : 5 : 4
- Total surface area (TSA) = 33,300 cm2
- Let l = 6k, b = 5k, h = 4k for some k > 0
Concept / Approach:
- Cuboid TSA = 2 * (l*b + b*h + h*l)
- Substitute proportional edges, solve for k, then compute actual dimensions.
Step-by-Step Solution:
Let l = 6k, b = 5k, h = 4k.TSA = 2 * (6k*5k + 5k*4k + 6k*4k) = 2 * (30k^2 + 20k^2 + 24k^2) = 2 * 74k^2 = 148k^2.Set 148k^2 = 33300 ⇒ k^2 = 33300 / 148 = 225 ⇒ k = 15.Thus l = 6*15 = 90 cm, b = 5*15 = 75 cm, h = 4*15 = 60 cm.
Verification / Alternative check:
Compute TSA with 90, 75, 60: l*b=6750, b*h=4500, h*l=5400; sum=16650; TSA=2*16650=33300 cm2, matches given.
Why Other Options Are Wrong:
- 90, 85, 60: Breaks the ratio 6 : 5 : 4.
- 85, 75, 60: Not in a constant multiple of 6 : 5 : 4.
- 90, 75, 70: Violates the 4-part for height.
Common Pitfalls:
- Forgetting to multiply all three pairwise products inside TSA.
- Using perimeter-like formula instead of surface area.
- Taking k = √(33300/148) but rounding incorrectly; here it is exactly 15.
Final Answer:
90, 75, 60
Discussion & Comments