To solve these problems, we will systematically use the measurements provided for the triangular prism:

(a) Find the lateral surface area of the prism (in square inches).

The lateral surface area of the prism includes the areas of the three rectangular faces (front, left, and right).

Calculations:

1. Front Face:
   Area = height × width = $10.0 \, \text{in} \times 17.0 \, \text{in} = 170.0 \, \text{in}^2$

2. Left Face:
   Area = height × width = $10.0 \, \text{in} \times 15.0 \, \text{in} = 150.0 \, \text{in}^2$

3. Right Face:
   Area = height × width = $10.0 \, \text{in} \times 12.0 \, \text{in} = 120.0 \, \text{in}^2$

Lateral Surface Area:
$170.0 + 150.0 + 120.0 = 440.0 \, \text{in}^2$

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(b) Find the total surface area of the prism (in square inches).

The total surface area includes the lateral surface area and the areas of the two triangular bases.

Step 1: Calculate the area of one triangular base.

The triangle's base is $17.0 \, \text{in}$, and its height is $10.3 \, \text{in}$ (given the line segment forming a right angle).

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 17.0 \, \text{in} \times 10.3 \, \text{in} = 87.55 \, \text{in}^2$

There are two triangular bases, so their combined area is: $2 \times 87.55 = 175.1 \, \text{in}^2$

Step 2: Add the lateral surface area.

$\text{Total Surface Area} = \text{Lateral Surface Area} + \text{Triangular Bases' Area}$ $\text{Total Surface Area} = 440.0 + 175.1 = 615.1 \, \text{in}^2$

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(c) Find the volume of the prism (in cubic inches).

The volume of a prism is calculated as: $\text{Volume} = \text{Base Area} \times \text{Height}$ Where:

• Base Area = $87.55 \, \text{in}^2$ (from part (b)),
• Height = $10.0 \, \text{in}$ (the perpendicular distance between the triangular bases).

$\text{Volume} = 87.55 \, \text{in}^2 \times 10.0 \, \text{in} = 875.5 \, \text{in}^3$

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Final Answers:

(a) Lateral Surface Area: 440.0 in²
(b) Total Surface Area: 615.1 in²
(c) Volume: 875.5 in³

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Would you like further explanations for any of the steps? Here are related questions for further exploration:

1. How is the height of the triangle determined from the given right angle?
2. How can the lateral surface area formula be generalized for any prism?
3. What is the significance of using precise measurements for surface area and volume calculations?
4. How do changes in the dimensions of the prism affect the total surface area and volume?
5. Can this formula be applied to a cylinder or a different prism shape?

Tip: Always double-check the right angle in your triangles, as it simplifies area and height calculations significantly!