Question
Marcus found a toy that is in the shape of a rectangular pyramid and is made out of plastic. The rectangular base has a length of 4 cm and a width of 5 cm. The height of the pyramid is 6 cm. Marcus weighs the toy and finds that it is 25 grams. Find the density of the toy.
Answers
GPT-4o mini
To find the density of the toy, we need to calculate its volume and then use the formula for density, which is given by:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
### Step 1: Calculate the Volume of the Rectangular Pyramid
The formula for the volume \( V \) of a rectangular pyramid is:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
First, we need to find the area of the base. The base is a rectangle with a length of 4 cm and a width of 5 cm.
\[
\text{Base Area} = \text{length} \times \text{width} = 4 \, \text{cm} \times 5 \, \text{cm} = 20 \, \text{cm}^2
\]
Now we can find the volume:
\[
V = \frac{1}{3} \times 20 \, \text{cm}^2 \times 6 \, \text{cm}
\]
\[
V = \frac{1}{3} \times 120 \, \text{cm}^3 = 40 \, \text{cm}^3
\]
### Step 2: Calculate the Density
Now that we have the mass of the toy (25 grams) and the volume (40 cm³), we can calculate the density:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{25 \, \text{g}}{40 \, \text{cm}^3}
\]
To simplify this, divide both the numerator and the denominator by 5:
\[
\text{Density} = \frac{25 \div 5}{40 \div 5} = \frac{5}{8} \, \text{g/cm}^3
\]
### Final Answer
Thus, the density of the toy is:
\[
\text{Density} = \frac{5}{8} \, \text{g/cm}^3 \quad \text{or} \quad 0.625 \, \text{g/cm}^3
\]
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
### Step 1: Calculate the Volume of the Rectangular Pyramid
The formula for the volume \( V \) of a rectangular pyramid is:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
First, we need to find the area of the base. The base is a rectangle with a length of 4 cm and a width of 5 cm.
\[
\text{Base Area} = \text{length} \times \text{width} = 4 \, \text{cm} \times 5 \, \text{cm} = 20 \, \text{cm}^2
\]
Now we can find the volume:
\[
V = \frac{1}{3} \times 20 \, \text{cm}^2 \times 6 \, \text{cm}
\]
\[
V = \frac{1}{3} \times 120 \, \text{cm}^3 = 40 \, \text{cm}^3
\]
### Step 2: Calculate the Density
Now that we have the mass of the toy (25 grams) and the volume (40 cm³), we can calculate the density:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{25 \, \text{g}}{40 \, \text{cm}^3}
\]
To simplify this, divide both the numerator and the denominator by 5:
\[
\text{Density} = \frac{25 \div 5}{40 \div 5} = \frac{5}{8} \, \text{g/cm}^3
\]
### Final Answer
Thus, the density of the toy is:
\[
\text{Density} = \frac{5}{8} \, \text{g/cm}^3 \quad \text{or} \quad 0.625 \, \text{g/cm}^3
\]