Question

A solid oblique cone with a slant length of 17 units is placed inside an empty cylinder with a congruent base of radius 8 units and a height of 15 units. What is the unfilled volume inside the cylinder? 320π cubic units 597π cubic units 640π cubic units 725π cubic units

Answer 1

Step $1$

Find the volume of a cylinder

we know that

volume of a cylinder is equal to

$V1=\pi r^{2} h$

in this problem

$r=8\ units \\ h=15\ units$

Substitute

$V1=\pi 8^{2} 15$

$V1=960\pi\ units^{3}$

Step $2$

Find the volume of a cone

we know that

volume of a cone is equal to

$V2=\frac{1}{3}\pi r^{2} h$

in this problem

$r=8\ units \\ l=17\ units \\ h=?$

applying the Pythagorean Theorem find the value of h

$h^{2} =l^{2}-r^{2}\\ h^{2} =17^{2}-8^{2}\\ h^{2}=225\\ h=15\ units$

$V2=\frac{1}{3}\pi 8^{2} 15\\ \\ V2=320\pi \ units^{3}$

Step $3$

Find the unfilled volume inside the cylinder

$V1-V2=960\pi -320\pi =640\pi \ units^{3}$

therefore

the answer is

the unfilled volume inside the cylinder is equal to $640\pi \ units^{3}$

Answer 2

The unfilled volume inside the cylinder is 640π unit³

Volume

Volume shows the amount of space enclosed by a closed surface. It is the amount of space that an object occupies, or that is enclosed within a container.

The volume of a cylinder = πr²h

where r is the radius,  h is the height.

r = 8., h = 15 feet, hence:

Volume = π(8)²(15) = 960π unit³

For the cone where l is the slant,  h is the height.

The volume of a cone = $\frac{1}{3}\pi r^2\sqrt{l^2*r^2}= \frac{1}{3}\pi (8)^2\sqrt{17^2*8^2}=320\pi \ unit^3$

Unfilled volume = 960π - 320 π = 640π unit³

The unfilled volume inside the cylinder is 640π unit³

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