Question

find the volume of:
a

b

c

d

e

f

Explanation:

Part a (Rectangular Prism)

Step1: Recall Volume Formula

The volume \( V \) of a rectangular prism is given by \( V = l \times w \times h \), where \( l \) is length, \( w \) is width, and \( h \) is height.

Step2: Substitute Values

Here, \( l = 4.2 \, \text{cm} \), \( w = 2.6 \, \text{cm} \), \( h = 2.3 \, \text{cm} \).
\( V = 4.2 \times 2.6 \times 2.3 \)
First, \( 4.2 \times 2.6 = 10.92 \)
Then, \( 10.92 \times 2.3 = 25.116 \, \text{cm}^3 \)

Part b (Triangular Prism)

Step1: Recall Volume Formula

The volume \( V \) of a triangular prism is \( V = \frac{1}{2} \times b \times h_{\text{tri}} \times l_{\text{prism}} \), where \( b \) is the base of the triangle, \( h_{\text{tri}} \) is the height of the triangle, and \( l_{\text{prism}} \) is the length of the prism.

Step2: Substitute Values

Here, \( b = 12.9 \, \text{cm} \), \( h_{\text{tri}} = 13.8 \, \text{cm} \), \( l_{\text{prism}} = 8.6 \, \text{cm} \).
\( V = \frac{1}{2} \times 12.9 \times 13.8 \times 8.6 \)
First, \( \frac{1}{2} \times 12.9 \times 13.8 = 6.45 \times 13.8 = 89.01 \)
Then, \( 89.01 \times 8.6 = 765.486 \, \text{cm}^3 \)

Part c (Cylinder)

Step1: Recall Volume Formula

The volume \( V \) of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Step2: Substitute Values

Here, \( r = 8.2 \, \text{cm} \), \( h = 13.9 \, \text{cm} \).
\( V = \pi \times (8.2)^2 \times 13.9 \)
\( (8.2)^2 = 67.24 \)
\( V = \pi \times 67.24 \times 13.9 \approx 3.1416 \times 67.24 \times 13.9 \approx 2972.47 \, \text{cm}^3 \) (using \( \pi \approx 3.1416 \))

Part d (Sphere)

Step1: Recall Volume Formula

The volume \( V \) of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius.

Step2: Substitute Values

Here, \( r = 4.8 \, \text{cm} \).
\( V = \frac{4}{3} \pi (4.8)^3 \)
\( (4.8)^3 = 4.8 \times 4.8 \times 4.8 = 110.592 \)
\( V = \frac{4}{3} \pi \times 110.592 \approx \frac{4}{3} \times 3.1416 \times 110.592 \approx 463.247 \, \text{cm}^3 \) (using \( \pi \approx 3.1416 \))

Part e (Pyramid)

Step1: Recall Volume Formula

The volume \( V \) of a pyramid is \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). The base is a right triangle, so base area \( = \frac{1}{2} \times b \times w \), where \( b = 2.8 \, \text{cm} \), \( w = 1.7 \, \text{cm} \), and height of the pyramid \( h = 2.9 \, \text{cm} \).

Step2: Substitute Values

Base Area \( = \frac{1}{2} \times 2.8 \times 1.7 = 2.38 \, \text{cm}^2 \)
\( V = \frac{1}{3} \times 2.38 \times 2.9 \approx \frac{1}{3} \times 6.902 \approx 2.301 \, \text{cm}^3 \)

Part f (Sphere)

Step1: Recall Volume Formula

The volume \( V \) of a sphere is \( V = \frac{4}{3} \pi r^3 \). The diameter is \( 3.7 \, \text{cm} \), so radius \( r = \frac{3.7}{2} = 1.85 \, \text{cm} \).

Step2: Substitute Values

\( V = \frac{4}{3} \pi (1.85)^3 \)
\( (1.85)^3 = 1.85 \times 1.85 \times 1.85 \approx 6.329 \)
\( V = \frac{4}{3} \pi \times 6.329 \approx \frac{4}{3} \times 3.1416 \times 6.329 \approx 26.53 \, \text{cm}^3 \) (using \( \pi \approx 3.1416 \))

Answer:

s:
a. \( \boldsymbol{25.116 \, \text{cm}^3} \)
b. \( \boldsymbol{765.486 \, \text{cm}^3} \)
c. \( \boldsymbol{\approx 2972.47 \, \text{cm}^3} \) (or exact form \( 8.2^2 \times 13.9 \pi \))
d. \( \boldsymbol{\approx 463.25 \, \text{cm}^3} \) (or exact form \( \frac{4}{3} \pi (4.8)^3 \))
e. \( \boldsymbol{\approx 2.30 \, \text{cm}^3} \) (or more precise \( \approx 2.301 \, \text{cm}^3 \))
f. \( \boldsymbol{\approx 26.53 \, \text{cm}^3} \) (or exact form \( \frac{4}{3} \pi (1.85)^3 \))