To solve this problem, we need to calculate the surface area of the prism in order to determine how much glitter Leon will need.

Step 1: Identify the shape of the prism.

The prism appears to have a triangular base, and we are given the following dimensions:

• The two slant heights of the triangle: $19 \, \text{m}$ and $27 \, \text{m}$
• The height of the triangular face: $10 \, \text{m}$
• The base of the triangular face: $23 \, \text{m}$
• The length of the prism: $12 \, \text{m}$

Step 2: Calculate the area of the triangular faces.

The area of a triangle is given by the formula: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ For the triangular faces: $\text{Area of triangular face} = \frac{1}{2} \times 23 \, \text{m} \times 10 \, \text{m} = 115 \, \text{m}^2$ Since there are two triangular faces (front and back), the total area of the triangular faces is: $\text{Total area of triangular faces} = 115 \, \text{m}^2 \times 2 = 230 \, \text{m}^2$

Step 3: Calculate the area of the rectangular faces.

Each side of the prism is a rectangle, and we can calculate the area of each of the three rectangular faces:

• One rectangular face has dimensions $23 \, \text{m} \times 12 \, \text{m}$, so: $\text{Area of first rectangular face} = 23 \times 12 = 276 \, \text{m}^2$
• The other two rectangular faces have dimensions of $27 \, \text{m} \times 12 \, \text{m}$ and $19 \, \text{m} \times 12 \, \text{m}$, so: $\text{Area of second rectangular face} = 27 \times 12 = 324 \, \text{m}^2$ $\text{Area of third rectangular face} = 19 \times 12 = 228 \, \text{m}^2$

Step 4: Calculate the total surface area.

Now, sum the areas of all the faces: $\text{Total surface area} = 230 \, \text{m}^2 + 276 \, \text{m}^2 + 324 \, \text{m}^2 + 228 \, \text{m}^2$ $\text{Total surface area} = 1058 \, \text{m}^2$

Step 5: Calculate the amount of glitter needed.

Since 70 grams of glitter are needed for each square meter: $\text{Total glitter} = 70 \, \text{g/m}^2 \times 1058 \, \text{m}^2 = 74060 \, \text{g}$ Thus, Leon will need 74.06 kg of glitter to cover the prism.

Would you like more details on any step or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How would the glitter requirement change if the base of the triangular face was doubled?
2. What is the total cost if glitter costs $5 per kilogram?
3. How would you calculate the volume of this triangular prism?
4. What happens if the height of the triangular face increases by 5 meters?
5. How can we generalize the glitter formula for any prism?

Tip: Always double-check your units when calculating area and volume to avoid mistakes!