Math Problem Statement

Work out the volume of these compound shapes.

Solution

Let's calculate the volumes of the two compound shapes (a and b) given in the image.

Shape a

Shape aa is made up of two rectangular prisms:

  1. Bottom section:

    • Length: 15m15 \, m
    • Width: 6m6 \, m
    • Height: 5m5 \, m

    Volume of bottom section: V1=15×6×5=450m3V_1 = 15 \times 6 \times 5 = 450 \, m^3

  2. Top section:

    • Length: 8m8 \, m
    • Width: 6m6 \, m
    • Height: 125=7m12 - 5 = 7 \, m

    Volume of top section: V2=8×6×7=336m3V_2 = 8 \times 6 \times 7 = 336 \, m^3

The total volume of shape aa is: Va=V1+V2=450+336=786m3V_a = V_1 + V_2 = 450 + 336 = 786 \, m^3

Shape b

Shape bb is also made up of two rectangular prisms:

  1. Top section:

    • Length: 20mm20 \, mm
    • Width: 7mm7 \, mm
    • Depth: 9mm9 \, mm

    Volume of top section: V3=20×7×9=1260mm3V_3 = 20 \times 7 \times 9 = 1260 \, mm^3

  2. Bottom section:

    • Length: 6mm6 \, mm
    • Width: 7mm7 \, mm
    • Depth: 3mm3 \, mm

    Volume of bottom section: V4=6×7×3=126mm3V_4 = 6 \times 7 \times 3 = 126 \, mm^3

The total volume of shape bb is: Vb=V3+V4=1260+126=1386mm3V_b = V_3 + V_4 = 1260 + 126 = 1386 \, mm^3

Final Results

  • Volume of shape aa: 786m3786 \, m^3
  • Volume of shape bb: 1386mm31386 \, mm^3

Would you like more details or have any questions?

Related Questions:

  1. How would the volumes change if shape aa had a different height for the top section?
  2. What is the volume of shape bb in cubic centimeters?
  3. How can we calculate the surface area of these compound shapes?
  4. If shape aa was scaled down by half, what would be its new volume?
  5. What are the differences in calculating volume for other 3D shapes like cylinders or spheres?

Tip:

Always double-check that all measurements are in the same units when calculating volumes to avoid errors.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Volume Calculation
Rectangular Prisms

Formulas

Volume of rectangular prism = length × width × height

Theorems

-

Suitable Grade Level

Grades 7-9