What is the whole surface area of the frustum?

Given:

Top diameter of the frustum (d₁) = 2k

Bottom diameter of the frustum (d₂) = 2.5k

Height of the frustum (h) = k

Formula used:

Radius of top (r₁) = d₁ / 2

Radius of bottom (r₂) = d₂ / 2

Slant height of frustum (l) = \(\sqrt{h^2 + (r_2 - r_1)^2}\)

Whole Surface Area of Frustum = Area of top base + Area of bottom base + Lateral Surface Area

Area of top base = \(\pi r_1^2\)

Area of bottom base = \(\pi r_2^2\)

Lateral Surface Area = \(\pi(r_1 + r_2)l\)

Calculations:

Calculate radii:

r₁ = 2k / 2 = k

r₂ = 2.5k / 2 = 1.25k

Calculate the difference in radii:

r₂ - r₁ = 1.25k - k = 0.25k

Calculate the slant height (l):

l = \(\sqrt{k^2 + (0.25k)^2}\)

⇒ l = \(\sqrt{k^2 + 0.0625k^2}\)

⇒ l = \(\sqrt{1.0625k^2}\)

⇒ l = k\(\sqrt{1.0625}\)

Now, calculate the surface areas:

Area of top base = \(\pi r_1^2\) = \(\pi (k)^2\) = \(\pi k^2\)

Area of bottom base = \(\pi r_2^2\) = \(\pi (1.25k)^2\) = \(\pi (1.5625k^2)\) = 1.5625\(\pi k^2\)

Lateral Surface Area = \(\pi(r_1 + r_2)l\)

⇒ Lateral Surface Area = \(\pi(k + 1.25k)k\sqrt{1.0625}\)

⇒ Lateral Surface Area = \(\pi(2.25k)k\sqrt{1.0625}\)

⇒ Lateral Surface Area = 2.25\(\pi k^2\sqrt{1.0625}\)

Whole Surface Area = \(\pi k^2\) + 1.5625\(\pi k^2\) + 2.25\(\pi k^2\sqrt{1.0625}\)

⇒ Whole Surface Area = \(\pi k^2 (1 + 1.5625 + 2.25\sqrt{1.0625})\)

⇒ Whole Surface Area = \(\pi k^2 (2.5625 + 2.25 \times 1.030776...)\)

⇒ Whole Surface Area = \(\pi k^2 (2.5625 + 2.319246...)\)

⇒ Whole Surface Area = \(\pi k^2 (4.881746...)\)

Let's use the fraction form for sqrt(1.0625) for precision:

1.0625 = 10625 / 10000 = 425 / 400 = 85 / 80 = 17/16 (Error in calculation here, 10625/10000 = 17/16 was for 1.0625, but 0.25k is (1/4)k)

0.25k = (1/4)k

(0.25k)² = (1/16)k²

l = \(\sqrt{k^2 + \frac{1}{16}k^2}\) = \(\sqrt{\frac{16k^2 + k^2}{16}}\) = \(\sqrt{\frac{17k^2}{16}}\) = \(\frac{k\sqrt{17}}{4}\)

Lateral Surface Area = \(\pi(k + 1.25k)\frac{k\sqrt{17}}{4}\)

⇒ Lateral Surface Area = \(\pi(2.25k)\frac{k\sqrt{17}}{4}\)

⇒ Lateral Surface Area = \(\pi(\frac{9}{4}k)\frac{k\sqrt{17}}{4}\)

⇒ Lateral Surface Area = \(\frac{9\pi k^2\sqrt{17}}{16}\)

Whole Surface Area = \(\pi k^2\) + \(\pi (\frac{5}{4}k)^2\) + \(\frac{9\pi k^2\sqrt{17}}{16}\)

⇒ Whole Surface Area = \(\pi k^2\) + \(\frac{25\pi k^2}{16}\) + \(\frac{9\pi k^2\sqrt{17}}{16}\)

⇒ Whole Surface Area = \(\pi k^2 \left(1 + \frac{25}{16} + \frac{9\sqrt{17}}{16}\right)\)

⇒ Whole Surface Area = \(\pi k^2 \left(\frac{16}{16} + \frac{25}{16} + \frac{9\sqrt{17}}{16}\right)\)

⇒ Whole Surface Area = \(\frac{\pi k^2}{16} (16 + 25 + 9\sqrt{17})\)

⇒ Whole Surface Area = \(\frac{\pi k^2}{16} (41 + 9\sqrt{17})\)

Using \(\sqrt{17} \approx 4.123\)

⇒ Whole Surface Area = \(\frac{\pi k^2}{16} (41 + 9 \times 4.123)\)

⇒ Whole Surface Area = \(\frac{\pi k^2}{16} (41 + 37.107)\)

⇒ Whole Surface Area = \(\frac{\pi k^2}{16} (78.107)\)

⇒ Whole Surface Area ≈ 4.8816 \(\pi k^2\)

The exact expression is \(\frac{\pi k^2}{16} (41 + 9\sqrt{17})\)

∴ The whole surface area of the frustum is \(\frac{\pi k^2}{16} (41 + 9\sqrt{17})\).