Question
A rectangular prism with a length of 8 centimeters, width of 15 centimeters, and an unknown height has a mass of 1,260 grams and a density of 1.2 g/cm3
1.2
g/cm
3
. Find the height of the prism.
1.2
g/cm
3
. Find the height of the prism.
Answers
GPT-4o mini
To find the height of the rectangular prism, we can use the relationship between mass, density, and volume, given by the formula:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
First, we need to find the volume of the rectangular prism. The volume \( V \) of a rectangular prism is calculated using the formula:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Given that the length is 8 cm and the width is 15 cm, we can express the volume in terms of the height \( h \):
\[
V = 8 \, \text{cm} \times 15 \, \text{cm} \times h = 120h \, \text{cm}^3
\]
Next, we can use the density formula to express mass in terms of density and volume:
\[
\text{Mass} = \text{Density} \times \text{Volume}
\]
We know that the mass of the prism is 1260 grams and the density is 1.2 g/cm³. Plugging in the values, we have:
\[
1260 \, \text{g} = 1.2 \, \text{g/cm}^3 \times V
\]
Now, substituting \( V = 120h \, \text{cm}^3 \) into the equation:
\[
1260 = 1.2 \times 120h
\]
Calculating \( 1.2 \times 120 \):
\[
1.2 \times 120 = 144
\]
So the equation simplifies to:
\[
1260 = 144h
\]
To find \( h \), we divide both sides by 144:
\[
h = \frac{1260}{144}
\]
Calculating \( h \):
\[
h = 8.75 \, \text{cm}
\]
Thus, the height of the rectangular prism is \( \mathbf{8.75 \, cm} \).
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
First, we need to find the volume of the rectangular prism. The volume \( V \) of a rectangular prism is calculated using the formula:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Given that the length is 8 cm and the width is 15 cm, we can express the volume in terms of the height \( h \):
\[
V = 8 \, \text{cm} \times 15 \, \text{cm} \times h = 120h \, \text{cm}^3
\]
Next, we can use the density formula to express mass in terms of density and volume:
\[
\text{Mass} = \text{Density} \times \text{Volume}
\]
We know that the mass of the prism is 1260 grams and the density is 1.2 g/cm³. Plugging in the values, we have:
\[
1260 \, \text{g} = 1.2 \, \text{g/cm}^3 \times V
\]
Now, substituting \( V = 120h \, \text{cm}^3 \) into the equation:
\[
1260 = 1.2 \times 120h
\]
Calculating \( 1.2 \times 120 \):
\[
1.2 \times 120 = 144
\]
So the equation simplifies to:
\[
1260 = 144h
\]
To find \( h \), we divide both sides by 144:
\[
h = \frac{1260}{144}
\]
Calculating \( h \):
\[
h = 8.75 \, \text{cm}
\]
Thus, the height of the rectangular prism is \( \mathbf{8.75 \, cm} \).