A right circular cylinder and a right circular cone have equal volumes. The radius of the … — SAT

A right circular cylinder and a right circular cone have equal volumes. The radius of the cylinder's base is $3$ times the radius of the cone's base. If the height of the cone is $k$ times the height of the cylinder, what is the value of $k$ ?

Cevap: 27

Cevap

The volume of a cylinder is

V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}

and the volume of a cone is

V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}

. Let the base radius of the cone be

r

and the height of the cylinder be

h

. According to the problem, the base radius of the cylinder is

3r

and the height of the cone is

kh

. Setting the volumes equal to each other gives

\pi (3r)^2 h = \frac{1}{3} \pi r^2 (kh)

. Squaring the term in parentheses simplifies the equation to

9 \pi r^2 h = \frac{k}{3} \pi r^2 h

. Dividing both sides by the common term

\pi r^2 h

yields

9 = \frac{k}{3}

. Multiplying both sides by 3 results in

k = 27

Adım Adım Çözüm

State the standard volume formulas for a right circular cylinder and a right circular cone.

V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}

and

V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}

To establish the mathematical equations governing the volume of each solid.

Define variables for the cone's radius (

r

) and the cylinder's height (

h

), then write the cylinder's radius and the cone's height using the given relationships.

r_{\text{cylinder}} = 3r

r_{\text{cone}} = r

h_{\text{cylinder}} = h

, and

h_{\text{cone}} = kh

To express all variables in terms of

r

h

, and the constant

k

so they can be compared directly.

Substitute the expressions into the volume formulas and set the two volumes equal to each other.

\pi (3r)^2 h = \frac{1}{3} \pi r^2 (kh)

The problem states that the cylinder and the cone have equal volumes.

Simplify the equation by squaring the cylinder's radius and dividing both sides by the common factors.

9 \pi r^2 h = \frac{k}{3} \pi r^2 h \implies 9 = \frac{k}{3} \implies k = 27

To isolate and solve for the constant

k

Anahtar Kavram

Relating the volumes of geometric solids through algebraic substitution and dimension scaling.

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