Question

The Germany historical society has just acquired a castle on the Rhine River and wishes to turn it into a museum. But for it to be inviting to guests, they must hear the castle and place a new layer of plaster around the outside. If one commercial heater can heat about 16,000 cubic feet, how many heaters will the Society need to purchase? How many square feet of plaster will they need?

Answer 1

`\large \boxed{\text{36 heaters; 31 816 ft}^(3)}`

Explanation:

Assume the dimensions of the castle are as in the diagram below.

There are four structures to consider:

• A rectangular prism
• A rectangular pyramid
• A cylinder
• A cone

1. Number of heaters

The total volume will have to be heated for the comfort of the guests.

(a) Volume of cylinder

d = 16 ft, so

r = 8 ft

V = πr²h = π × (8 ft)² × 60 ft = 12 063 ft³

(b) Volume of cone

V = ⅓πr²h = ⅓π × (8 ft)² × 15 ft = 1005 ft³

(c) Volume of rectangular prism

V = lwh = 198 ft × 42 ft × 60 ft = 498 960 ft³

(d) Volume of rectangular pyramid

(i) Height of pyramid

AC² + BC² = AB²

AC² + (99 ft)² =(101 ft)²

AC² + 9801 ft² =10 201 ft²

AC² = 400 ft²

h = AC = 20 ft

(ii) Volume of pyramid

V = ⅓Bh = ⅓ × 198 ft × 42 ft × 20 ft = 55 440 ft³

(e) Total volume

V = 12 064 ft³ + 1005 ft³ + 498 960 ft³ +55 440 ft³ = 567 469 ft³

(f) Number of heaters

`\text{No. of heaters} = \text{567 469 ft}^(3) * \frac{\text{1 heater}}{\text{16 000 ft}^(3)} = \text{35.5 heaters} \Rightarrow \textbf{36 heaters}\\\\\text{The Society will need to purchase $\large \boxed{\textbf{36 heaters}}$}`

2. Amount of plaster

Only the vertical surfaces of the castle must be plastered. The roofs are probably of slate.

We need to consider only the lateral surface areas of the rectangular prism and the cylinder.

(a) Lateral area of rectangular prism

LA = 2lh +2wh = 2 × 198 ft × 60 ft + 2 × 42 ft × 60 ft = 23 760 ft² + 5040 ft² = 28 800 ft²

(b) Lateral area of cylinder

LA = 2πrh = 2π × 8 ft × 60 ft = 3016 ft²

(c) Amount of plaster

Plaster = 28 800 ft² + 3016 ft ² = 31 816 ft²