Math / Geometry

QuestionKaitlin's company makes solid spherical metal balls for various industrial uses. A customer wants copper balls that have a diameter of 3 in. If Kaitlin must make 90 of these balls, how much copper will she need?
If necessary, refer to the list of geometry formulas. For your calculations, do not round any intermediate steps, and use the $\pi$ button on the ALEKS calculator. Round your answer to the nearest hundredth. $\square$ ${ }^{3}$

Studdy Solution

STEP 1

1. The metal balls are perfect spheres.
2. The diameter of each sphere is $3$ inches.
3. Kaitlin needs to make $90$ spheres.
4. The volume of a sphere is calculated using the formula $V = \frac{4}{3} \pi r^3$ .

STEP 2

1. Determine the radius of the sphere.
2. Recall the formula for the volume of a sphere.
3. Calculate the volume of one sphere.
4. Calculate the total volume for 90 spheres.
5. Round the total volume to the nearest hundredth.

STEP 3

Determine the radius of the sphere:
The diameter of the sphere is $3$ inches, so the radius $r$ is half of the diameter:
$r = \frac{3}{2} = 1.5 \text{ inches}$

STEP 4

Recall the formula for the volume of a sphere:
$V = \frac{4}{3} \pi r^3$

STEP 5

Calculate the volume of one sphere using the radius:
$V = \frac{4}{3} \pi (1.5)^3$
Calculate $(1.5)^3$ :
$(1.5)^3 = 3.375$
Substitute back into the volume formula:
$V = \frac{4}{3} \pi \times 3.375$

STEP 6

Calculate the total volume for 90 spheres:
First, calculate the volume of one sphere:
$V = \frac{4}{3} \pi \times 3.375 \approx 14.13716694 \text{ cubic inches}$
Now, multiply by 90 to find the total volume:
$\text{Total Volume} = 90 \times 14.13716694$
$\text{Total Volume} \approx 1272.3440246 \text{ cubic inches}$

STEP 7

Round the total volume to the nearest hundredth:
$\text{Total Volume} \approx 1272.34 \text{ cubic inches}$
The total amount of copper needed is:
$\boxed{1272.34 \text{ cubic inches}}$

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Studdy Solution

STEP 1

1. The metal balls are perfect spheres.
2. The diameter of each sphere is $3$ inches.
3. Kaitlin needs to make $90$ spheres.
4. The volume of a sphere is calculated using the formula $V = \frac{4}{3} \pi r^3$ .

STEP 2

1. Determine the radius of the sphere.
2. Recall the formula for the volume of a sphere.
3. Calculate the volume of one sphere.
4. Calculate the total volume for 90 spheres.
5. Round the total volume to the nearest hundredth.

STEP 3

Determine the radius of the sphere:
The diameter of the sphere is $3$ inches, so the radius $r$ is half of the diameter:
$r = \frac{3}{2} = 1.5 \text{ inches}$

STEP 4

Recall the formula for the volume of a sphere:
$V = \frac{4}{3} \pi r^3$

STEP 5

Calculate the volume of one sphere using the radius:
$V = \frac{4}{3} \pi (1.5)^3$
Calculate $(1.5)^3$ :
$(1.5)^3 = 3.375$
Substitute back into the volume formula:
$V = \frac{4}{3} \pi \times 3.375$

STEP 6

STEP 7

Round the total volume to the nearest hundredth:
$\text{Total Volume} \approx 1272.34 \text{ cubic inches}$
The total amount of copper needed is:
$\boxed{1272.34 \text{ cubic inches}}$

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Studdy Solution

STEP 1

1. The metal balls are perfect spheres.2. The diameter of each sphere is 3 3 3 inches.3. Kaitlin needs to make 90 90 90 spheres.4. The volume of a sphere is calculated using the formula V=43πr3 V = \frac{4}{3} \pi r^3 V=34​πr3.

STEP 2

1. Determine the radius of the sphere.2. Recall the formula for the volume of a sphere.3. Calculate the volume of one sphere.4. Calculate the total volume for 90 spheres.5. Round the total volume to the nearest hundredth.

STEP 3

Determine the radius of the sphere:The diameter of the sphere is 3 3 3 inches, so the radius r r r is half of the diameter:r=32=1.5 inches r = \frac{3}{2} = 1.5 \text{ inches} r=23​=1.5 inches

STEP 4

Recall the formula for the volume of a sphere:V=43πr3 V = \frac{4}{3} \pi r^3 V=34​πr3

STEP 5

Calculate the volume of one sphere using the radius:V=43π(1.5)3 V = \frac{4}{3} \pi (1.5)^3 V=34​π(1.5)3Calculate (1.5)3 (1.5)^3 (1.5)3:(1.5)3=3.375 (1.5)^3 = 3.375 (1.5)3=3.375Substitute back into the volume formula:V=43π×3.375 V = \frac{4}{3} \pi \times 3.375 V=34​π×3.375

STEP 6

STEP 7

Round the total volume to the nearest hundredth:Total Volume≈1272.34 cubic inches \text{Total Volume} \approx 1272.34 \text{ cubic inches} Total Volume≈1272.34 cubic inches The total amount of copper needed is:1272.34 cubic inches \boxed{1272.34 \text{ cubic inches}} 1272.34 cubic inches​

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1. The metal balls are perfect spheres.
2. The diameter of each sphere is $3$ inches.
3. Kaitlin needs to make $90$ spheres.
4. The volume of a sphere is calculated using the formula $V = \frac{4}{3} \pi r^3$ .

1. Determine the radius of the sphere.
2. Recall the formula for the volume of a sphere.
3. Calculate the volume of one sphere.
4. Calculate the total volume for 90 spheres.
5. Round the total volume to the nearest hundredth.

Determine the radius of the sphere:
The diameter of the sphere is $3$ inches, so the radius $r$ is half of the diameter:
$r = \frac{3}{2} = 1.5 \text{ inches}$

Recall the formula for the volume of a sphere:
$V = \frac{4}{3} \pi r^3$

Calculate the volume of one sphere using the radius:
$V = \frac{4}{3} \pi (1.5)^3$
Calculate $(1.5)^3$ :
$(1.5)^3 = 3.375$
Substitute back into the volume formula:
$V = \frac{4}{3} \pi \times 3.375$

Round the total volume to the nearest hundredth:
$\text{Total Volume} \approx 1272.34 \text{ cubic inches}$
The total amount of copper needed is:
$\boxed{1272.34 \text{ cubic inches}}$