Math Problem Statement

solve just simple working out

Solution

Let's solve each problem step-by-step.

Problem 7

We need to calculate the volume of the prism.

The prism's volume VV can be found using the formula: V=Area of cross-section×Length (or height)V = \text{Area of cross-section} \times \text{Length (or height)}

Given:

  • Area of cross-section = 42 m²
  • Length (or height) = 8 m

Calculation: V=42m2×8m=336m3V = 42 \, \text{m}^2 \times 8 \, \text{m} = 336 \, \text{m}^3

Answer for Problem 7: The volume of the prism is 336 m³.

Problem 8

We need to find the volume of each prism by first calculating the area of the cross-section.

Problem 8a

Shape: Triangular Prism

Dimensions given:

  • Base of triangle = 8 cm
  • Height of triangle = 6 cm
  • Length of prism = 10 cm

Steps:

  1. Calculate the area of the triangle cross-section using the formula: Area=12×Base×Height\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} Area=12×8cm×6cm=24cm2\text{Area} = \frac{1}{2} \times 8 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2

  2. Calculate the volume of the prism using the formula: V=Area of cross-section×LengthV = \text{Area of cross-section} \times \text{Length} V=24cm2×10cm=240cm3V = 24 \, \text{cm}^2 \times 10 \, \text{cm} = 240 \, \text{cm}^3

Answer for Problem 8a: The volume of the prism is 240 cm³.

Problem 8b

Shape: Trapezoidal Prism

Dimensions given:

  • Top base of trapezoid = 4 m
  • Bottom base of trapezoid = 8 m
  • Height of trapezoid = 5 m
  • Length of prism = 12 m

Steps:

  1. Calculate the area of the trapezoid cross-section using the formula: Area=12×(Base1+Base2)×Height\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} Area=12×(4m+8m)×5m\text{Area} = \frac{1}{2} \times (4 \, \text{m} + 8 \, \text{m}) \times 5 \, \text{m} Area=12×12m×5m=30m2\text{Area} = \frac{1}{2} \times 12 \, \text{m} \times 5 \, \text{m} = 30 \, \text{m}^2

  2. Calculate the volume of the prism using the formula: V=Area of cross-section×LengthV = \text{Area of cross-section} \times \text{Length} V=30m2×12m=360m3V = 30 \, \text{m}^2 \times 12 \, \text{m} = 360 \, \text{m}^3

Answer for Problem 8b: The volume of the prism is 360 m³.


Would you like more details on any of these steps, or do you have any questions?

Related Questions:

  1. How do you find the area of a circle and use it to calculate the volume of a cylindrical prism?
  2. What is the formula for finding the surface area of a prism?
  3. How does the volume formula change for prisms with different cross-sectional shapes?
  4. Can you explain how to find the volume of a pyramid and how it differs from finding the volume of a prism?
  5. How would you calculate the volume of a prism with a polygonal cross-section like a pentagon or hexagon?

Tip: When calculating volumes, always ensure your cross-sectional area is correctly calculated based on the specific shape of the prism.

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Math Problem Analysis

Mathematical Concepts

Geometry
Prisms
Volumes

Formulas

Volume of a prism

Theorems

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Suitable Grade Level

Grades 5-7