3D geometry – An order of mine (basic)

Concepts addressed: square pyramid volume, triangular prism volume, percentages.
Difficulty level: Basic.
An advanced version of An order of mine is also available.

Scenario:

You own a smithy in a small rural town some few miles away from some high mountains. Today seemed like just another lazy day, but suddenly a very short, bearded man enters your shop and greets you with a thick accent you don’t recognize. He sure is an odd client, but he soon attracts your attention with the big order he wants you to make.

1. The client would like you to prepare for him twenty-five pickaxes. The head of each pickaxe has on one side a right square pyramid, which is 9-inches high, with a 3-inch base edge, and on the other side a triangular prism, which has a height of 3 inches and whose base is a right triangle with legs that are 3- and 6-inches long. What volume of metal do you need to fulfil the order if each pickaxe needs additional three cubic inches of metal for the joint between the two parts?
1. The visitor offers you \$9,000 for the whole order. You buy the metal for \$5 per cubic inch, and every pickaxe needs a \$5 wooden handle. How much are you going to earn on this deal if you have to subtract 15% from the shop’s profit to pay the local girl that helps you?

Useful calculators:

Question 1 hints:

Hint 1
Try to sketch what the pickaxe head looks like. What parts does it consist of? Can you describe the sides of the shapes?
Hint 2
What is the volume of the pyramid part? And of the prism part? How many of each do we have to produce?
Hint 3
If we know the volume of all of the pyramid and prism parts, is that our final answer, or do we have to take something more into account?

Question 2 hints:

Hint 1
From the first question, we know how much metal we need, but how much will it cost us?
Hint 2
What is the cost of all of the components needed, i.e., the metal and the handles?
Hint 3
How much did the shop earn on the deal? How can we see how much of that goes to you?

Solutions:

Question 1
V = 1,425 in³
Question 2
earnings = \$1,487.50

Step-by-step solution:

Question 1
Let’s begin by calculating the volume of metal we need for a single pickaxe. The scenario tells us that its head has two parts: a right square pyramid, and a right triangle prism. The first one has a height, h_pyramid, of 9 in, and a base side, a, of 3 in. Therefore, the right rectangular pyramid calculator gives that its volume is
V_pyramid = (a² * h_pyramid) / 3 = ((3 in)² * 9 in) / 3 = 27 in³.

On the other hand, the prism has a right triangular base, with legs b = 3 in and c = 6 in. Observe that we can easily calculate the area of this triangle since b can be its height, and c its width (or vice versa). Moreover, the prism has a height, h_prism, of 3 in. Now we use the triangular prism calculator to get
V_prism = h_prism * (b * c) / 2 = 3 in * (3 in * 6 in) / 2 = 27 in³.

We are now ready to calculate how much metal is needed for a single pickaxe. Remember that each requires an additional 3 in³ for the joint. This means that:
V_pickaxe = V_pyramid + V_prism + 3 in³ = 27in³ + 27 in³ + 3 in³ = 57 in³.

Lastly, we multiply the volume needed for one pickaxe by the number the client ordered:
V = 25 * V_pickaxe = 25 * 57 in³ = 1,425 in³.

Question 2
Let us begin by calculating the costs of the whole order. We know from Question 1 how much metal we need, so now we can multiply that by the cost of a single cubic inch to get:
metal_cost = V * \$5 = 1,425 * \$5 = \$7,125.
Recall that each of the 25 pickaxes needs a \$5 handle. Therefore,
total_cost = metal_cost + 25 * \$5 = \$7,125 + \$125 = \$7,250.

This means that the shop has made \$9,000 – \$7,250 = \$1,750 on this order. Maybe it’s time to reconsider our area of work?

Anyway, the last thing to do is to take away 15% from this sum for the helper. This means that you’ll be left with 100% – 15% = 85% of the profit. The percentage calculator shows that:
earnings = 85% * \$1,750 = \$1,487,50.

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