Concepts addressed: cube volume, triangular prism volume, square pyramid volume, cylinder volume, percentages.
Recommended grade: 7th.
Difficulty level: Advanced.
A basic version of An order of mine is also available.
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 visitor wants 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 make all of the pyramids and prisms?
- In each pickaxe, the two parts of its head are joined by a joint. It has the form of a cube with a 3-inch-long edge with a hole right through two oposing faces. The hole is in the shape of a cylinder with base diameter of 2.5 in, and is then used to hold the handle. With the joints in mind, how much metal will you need for the whole order?
- The visitor offers you a pure gold nugget weighing 6 oz for the whole order. You buy the metal for $5 per cubic inch, and every pickaxe needs a $5 wooden handle. Suppose that gold is worth $1,700 per ounce. 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?
- Right rectangular pyramid calculator – https://www.omnicalculator.com/math/right-rectangular-pyramid
- Triangular prism calculator – https://www.omnicalculator.com/math/triangular-prism
- Cube calculator – https://www.omnicalculator.com/math/cube
- Volume of a cylinder calculator – https://www.omnicalculator.com/math/cylinder-volume
- Percentage calculator – https://www.omnicalculator.com/math/percentage
Question 1 hints:
Question 2 hints:
Question 3 hints:
Solutions (WARNING: depends on the current gold value, example for $1,658):
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. We can use the slope calculator to calculate the angle we want: it has legs b & c equal to 3 in and 6 in respectively. Observe that we can easily calculate the area of this triangle since b can be the triangle’s height, and c its width (or vice versa). Moreover, the prism has a height, h_prism, of 3 in. Now we can use the triangular prism calculator to get:
V_prism = h_prism * (b * c) / 2 = 3 in * (3 in * 6 in) / 2 = 27 in³.
Therefore, since we’ll be making 25 pickaxes, we need:
V₁ = 25 * (V_pyramid + V_prism) = 25 * (27 in³ + 27 in³) = 1,350 in³
With this we are ready to use the volume of a cylinder calculator:
V_cube = a³ = (3 in)³ = 27 in³,
V_hole = h * π * r² = 3 in * π * (1.25 in)² ≈ 14.72 in³.
Hence, the volume of a single joint is:
V_joint = V_cube – V_hole ≈ 27 in³ – 14.72 in³ = 12.28 in³.
Lastly, we need to calculate how much metal we need for all the pickaxe heads. To do this, let’s take the result of Question 1 and add the 25 joints:
V₂ = V₁ + 25 * V_joint ≈ 1,350 in³ + 25 * 12.28 in³ = 1,657 in³.
metal_cost = V₂ * $5 = 1,657 * $5 = $8,285.
Recall that each of the 25 pickaxes needs a $5 handle. This means that:
total_cost = metal_cost + 25 * $5 = $8,285 + $125 = $8,410.
Now let’s calculate the value of the 6-ounce gold nugget. If one ounce of pure gold is worth $1,700, then the shop has received 6 * $1,700 = $10,200 worth of gold. This means that the shop has made $10,200 – $8,410 = $1,790 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 gives that this is:
earnings = 85% * $1,790 ≈ $1,521.50.