Respuesta :
Answer:
1. The radius of the sphere is 10.61 in.
2. The density of the rectangular prism is or 0.29
3. The volume of the rubber shell is 443 in.³
The volume of the sphere is 265.96 in.³
Step-by-step explanation:
The formula for the volume of a sphere is presented as follows;
[tex]Volume \ of \ a \ sphere \ V = \frac{4}{3} \times \pi \times r^3[/tex]
Given that the volume = 5000 in.³, we have;
[tex]5000 = \frac{4}{3} \times \pi \times r^3[/tex]
[tex]\therefore r^3 = \frac{3}{4} \times \frac{5000}{\pi } = \frac{3750}{\pi }[/tex]
Hence;
[tex]\therefore r = \sqrt[3]{\frac{3750}{\pi }} = 10.61 \ in.[/tex]
The radius of the sphere = 10.61 in.
2. Volume of the rectangular prism = Length, l × Breadth, b × Height, h
Where:
l = 6 cm, b = 4 cm, and h = 5 cm, we have;
The volume, V of the rectangular prism = 6 × 4 × 5 = 120 cm³
The mass, m of the rectangular prism = 35 grams, therefore, since;
[tex]Density, D = \frac{Mass, \ m}{Volume, \ V}[/tex]
[tex]\therefore Density, D = \frac{35 \, g}{120 \, cm^3}= \frac{7}{24} \frac{g}{cm^3}[/tex]
The density of the rectangular prism = 7/24 g/cm³ or 0.29
3. Here we have the outer diameter of the rubber shell = 24 in.
Volume enclosed by the outer shell is found from the volume of a sphere = [tex]Volume \ of \ a \ sphere \ V = \frac{4}{3} \times \pi \times r^3[/tex]
Where:
r = Radius of the sphere = half the diameter = 24/2 = 12 in,
Therefore, volume of the ball outer shell = 4/3×π×12³ = 7238.2 in.³
Also the inner diameter of the rubber shell = 23.5 in, Therefore;
Volume enclosed by the inner shell = 4/3×π×(23.5/2)³ = 6795.2 in.³
The volume of the rubber shell = Volume of the ball outer shell - (Volume of the ball inner shell)
The volume of the rubber shell = 7238.2 in.³ - 6795.2 in.³ = 443.03 in.³ ≈ 443 in.³
The volume of the rubber shell ≈ 443 in.³
4. Here we have, the formula for the surface area, A of a sphere = 4·π·r²
Whereby, A = 200 in.², we have;
200 = 4·π·r²
Making r the subject of the above formula gives;
r = √(200/(4·π)) = 3.99 in.
Hence the volume, V of the sphere is found from [tex]V = \frac{4}{3} \times \pi \times r^3[/tex];
Hence we have;
[tex]V = \frac{4}{3} \times \pi \times 3.99^3 = 265.96 \ in.^3[/tex]
The volume, V of the sphere = 265.96 in.³.
