2. a wooden stool is in the form of a frustum of a cone with slant edge 40cm, top diameter 30cm and bottom diameter 50cm.
(a) calculate the perpendicular height of the stool
d₁ = 30cm
r₁ = 30/2 =15 cm
bottom d₂ =50cm
r₂ = 25cm
sl = 40cm
con h = √40² - (25 - 15)² = √(1600 - 100) = √1500 ≈ 38.73 cm
(b) calculate the total surface area of the stool
a = π(15)² + (25)² + 40×(15 + 25)
= 3.142×(225 + 625 + 40×40)
= 3.142×(225 + 625 + 1600)
= 3.142×2450 ≈ 7696.9 cm²
(c) calculate the volume of wood used to make the stool in terms of π
v = (1/3)π×10×(15² + 25² + 15×25)
= (1/3)π×10×(225 + 625 + 375)
= (1/3)π×10×1225
= (12250/3)π ≈ 4083.33π cm³

Question

Accepted Answer

### Part (a): Calculate the perpendicular height of the stool
# Explanation:
## Step 1: Identify radii and slant height
The top diameter $ D_1 = 30 \, \text{cm} $, so top radius $ R_1=\frac{30}{2}=15 \, \text{cm} $. The bottom diameter (assuming the bottom radius $ R_2 $ is from the bottom diameter? Wait, the bottom diameter is 60 cm? Wait, the original problem: "top diameter 30cm and bottom diameter 60cm" (from the writing: $ D_2 = 60 \, \text{cm} $, so $ R_2=\frac{60}{2}=30 \, \text{cm} $? Wait, the handwritten has $ R_2 = 25 \, \text{cm} $? Wait, maybe a typo. Wait, the slant height $ l = 40 \, \text{cm} $. The formula for the height $ h $ of a frustum is $ h=\sqrt{l^{2}-(R_1 - R_2)^{2}} $ (if $ R_1 < R_2 $, else $ |R_1 - R_2| $). Wait, the top radius $ R_1 = 15 \, \text{cm} $ (30/2), bottom radius $ R_2 = 30 \, \text{cm} $ (60/2)? But the handwritten has $ R_2 = 25 \, \text{cm} $. Wait, maybe the bottom diameter is 50 cm? Wait, the original problem: "top diameter 30cm and bottom diameter 50cm"? Wait, the handwritten says $ R_2 = 25 \, \text{cm} $ (since 50/2=25). Then slant height $ l = 40 \, \text{cm} $. Then the height $ h=\sqrt{l^{2}-(R_2 - R_1)^{2}}=\sqrt{40^{2}-(25 - 15)^{2}}=\sqrt{1600 - 100}=\sqrt{1500}\approx38.73 \, \text{cm} $, which matches the handwritten. So:

## Step 1: Find radii
Top radius $ R_1=\frac{30}{2}=15 \, \text{cm} $, bottom radius $ R_2=\frac{50}{2}=25 \, \text{cm} $ (assuming bottom diameter 50 cm, since $ R_2 = 25 $). Slant height $ l = 40 \, \text{cm} $.

## Step 2: Apply frustum height formula
The height $ h $ of a frustum is given by $ h=\sqrt{l^{2}-(R_2 - R_1)^{2}} $ (since $ R_2 > R_1 $). Substitute values: $ h=\sqrt{40^{2}-(25 - 15)^{2}}=\sqrt{1600 - 100}=\sqrt{1500}\approx38.73 \, \text{cm} $.

# Answer (a): $ \approx38.73 \, \text{cm} $

### Part (b): Calculate total surface area of the stool
The total surface area (TSA) of a frustum of a cone is $ \pi(R_1^{2}+R_2^{2}+l(R_1 + R_2)) $, where $ l $ is slant height.

## Step 1: Identify values
$ R_1 = 15 \, \text{cm} $, $ R_2 = 25 \, \text{cm} $, $ l = 40 \, \text{cm} $, $ \pi\approx3.142 $.

## Step 2: Substitute into formula
$ \text{TSA}=\pi(15^{2}+25^{2}+40(15 + 25)) $

## Step 3: Calculate each term
$ 15^{2}=225 $, $ 25^{2}=625 $, $ 40(15 + 25)=40\times40 = 1600 $

## Step 4: Sum and multiply by $ \pi $
$ \text{TSA}=3.142\times(225 + 625 + 1600)=3.142\times2450\approx7696.9 \, \text{cm}^2 $, which matches the handwritten.

# Answer (b): $ \approx7696.9 \, \text{cm}^2 $

### Part (c): Calculate volume of wood used
The volume $ V $ of a frustum of a cone is $ V=\frac{1}{3}\pi h(R_1^{2}+R_2^{2}+R_1R_2) $, or alternatively, using the formula from the handwritten: $ V=\frac{1}{3}\pi\times10\times(16^{2}+26^{2}+16\times26) $? Wait, no, earlier we had $ R_1 = 15 $, $ R_2 = 25 $, $ h\approx38.73 $. But the handwritten uses $ V=\frac{1}{3}\pi\times10\times(16^{2}+26^{2}+16\times26) $? Wait, maybe a typo. Wait, the correct formula is $ V=\frac{1}{3}\pi h(R_1^{2}+R_2^{2}+R_1R_2) $. Using $ R_1 = 15 $, $ R_2 = 25 $, $ h\approx38.73 $:

## Step 1: Apply frustum volume formula
$ V=\frac{1}{3}\pi h(R_1^{2}+R_2^{2}+R_1R_2) $

## Step 2: Substitute values
$ h\approx38.73 $, $ R_1 = 15 $, $ R_2 = 25 $

$ V=\frac{1}{3}\pi\times38.73\times(15^{2}+25^{2}+15\times25) $

$ 15^{2}=225 $, $ 25^{2}=625 $, $ 15\times25 = 375 $

Sum: $ 225 + 625 + 375 = 1225 $

$ V=\frac{1}{3}\pi\times38.73\times1225\approx\frac{1}{3}\pi\times47444.25\approx15814.75\pi\approx49680 \, \text{cm}^3 $ (approximate). But the handwritten has a different approach, maybe using the original cone volume minus the smaller cone volume. Let the height of the larger cone be $ H $, smaller cone be $ H - h $. Since $ \frac{R_1}{R_2}=\frac{15}{25}=\frac{3}{5} $, so $ \frac{H - h}{H}=\frac{3}{5}\implies5(H - h)=3H\implies2H = 5h\implies H=\frac{5h}{2}\approx\frac{5\times38.73}{2}\approx96.825 \, \text{cm} $. Then volume of larger cone $ V_2=\frac{1}{3}\pi R_2^{2}H $, smaller cone $ V_1=\frac{1}{3}\pi R_1^{2}(H - h) $. Then $ V = V_2 - V_1=\frac{1}{3}\pi(25^{2}\times96.825 - 15^{2}\times(96.825 - 38.73)) $. But this is more complex. The handwritten formula seems to have a typo, but the key is using the frustum volume formula.

However, based on the handwritten, the final answer for (c) is "1015 cm" (probably $ 1015 \, \text{cm}^3 $ or a miscalculation, but the correct approach is using the frustum volume formula.

But since the problem is about a wooden stool (frustum), the volume is calculated using the frustum volume formula.

# Answer (a): $ \approx38.73 \, \text{cm} $
# Answer (b): $ \approx7697 \, \text{cm}^2 $ (rounded)
# Answer (c): Using $ V=\frac{1}{3}\pi h(R_1^{2}+R_2^{2}+R_1R_2) \approx \frac{1}{3}\times3.142\times38.73\times(225 + 625 + 375)\approx49680 \, \text{cm}^3 $ (approximate)

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QUESTION IMAGE

Question

Explanation:

Part (a): Calculate the perpendicular height of the stool

Step 1: Identify radii and slant height

Step 1: Find radii

Step 2: Apply frustum height formula

Answer:

Part (b): Calculate total surface area of the stool

Step 1: Identify values

Step 2: Substitute into formula

Step 3: Calculate each term

Step 4: Sum and multiply by \( \pi \)