A solid right circular cylinder has a radius of r centimeters and a height of 6 centimeter… — SAT

A solid right circular cylinder has a radius of $r$ centimeters and a height of $6$ centimeters. A solid sphere has a radius of $r$ centimeters. If the sum of the volumes of the cylinder and the sphere is equal to the volume of a right circular cylinder with a radius of $2r$ centimeters and a height of $2$ centimeters, what is the value of $r$ ?

Cevap: 1.5 centimeters

Cevap

The correct answer is 1.5 (or 3/2).

The correct answer is 1.5. By using the formulas for the volume of a cylinder (

V = \pi r^2 h

) and a sphere (

V = \frac{4}{3}\pi r^3

), the total volume of the first cylinder and the sphere is

6\pi r^2 + \frac{4}{3}\pi r^3

. The second cylinder has a volume of

\pi(2r)^2(2) = 8\pi r^2

. Setting them equal gives

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

. Subtracting

6\pi r^2

from both sides gives

\frac{4}{3}\pi r^3 = 2\pi r^2

. Dividing both sides by

\pi r^2

(since

r > 0

) simplifies to

\frac{4}{3}r = 2

, which solves to

r = 1.5

Adım Adım Çözüm

Express the volume of the first cylinder using the formula

V = \pi r^2 h

V_1 = 6\pi r^2

To represent the cylinder's volume in terms of its radius.

Express the volume of the sphere using the formula

V = \frac{4}{3}\pi r^3

V_2 = \frac{4}{3}\pi r^3

To represent the sphere's volume in terms of its radius.

Express the volume of the second cylinder, which has a radius of

2r

and a height of

2

V_3 = \pi (2r)^2(2) = 8\pi r^2

To represent the larger cylinder's volume in terms of

r

Set up the equation where the sum of the first cylinder and sphere volumes equals the second cylinder volume.

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

To relate the volumes as given in the problem statement.

Simplify the equation by subtracting

6\pi r^2

from both sides.

\frac{4}{3}\pi r^3 = 2\pi r^2

To isolate the cubic term on one side.

Divide both sides of the equation by

\pi r^2

since

r > 0

\frac{4}{3}r = 2

To reduce the polynomial equation to a linear equation.

Multiply both sides by

\frac{3}{4}

to solve for

r

r = 1.5

To find the final numerical value of the radius.

Anahtar Kavram

Volume of cylinders and spheres

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru