Concepts addressed: **square pyramid volume**, **triangular prism volume**, **percentages**.

Recommended grade: **7th**.

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.

- 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?

- 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:**

- Right rectangular pyramid calculator – https://www.omnicalculator.com/math/right-rectangular-pyramid
- Triangular prism calculator – https://www.omnicalculator.com/math/triangular-prism
- Percentage calculator – https://www.omnicalculator.com/math/percentage

**Question 1 hints:**

**Question 2 hints:**

**Solutions:**

**Step-by-step solution:**

**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 in ³ = 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 in ³ = 1,425 in³*.

**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*.

A bonus:what are you teaching next week? We'd love to prepare a scenario for you 🙂