A solid wooden block is shaped as a frustum of a right circular cone with a top base radiu… — JAMB UTME

A solid wooden block is shaped as a frustum of a right circular cone with a top base radius of $4\text{ cm}$ , a bottom base radius of $12\text{ cm}$ , and a vertical height of $15\text{ cm}$ . A cylindrical hole of radius $4\text{ cm}$ is drilled vertically through the center of the frustum from the top base straight down to the bottom base. What is the volume of the remaining wooden solid in $\text{cm}^3$ ?

$800\pi\text{ cm}^3$ Cevap
B
$2880\pi\text{ cm}^3$
C
$560\pi\text{ cm}^3$
D
$80\pi\text{ cm}^3$

Cevap

The volume of the remaining wooden solid is

800\pi\text{ cm}^3

The volume of a frustum of a cone with radii

R=12\text{ cm}

r=4\text{ cm}

, and height

h=15\text{ cm}

\frac{1}{3}\pi(15)(12^2 + 4^2 + 12 \times 4) = 1040\pi\text{ cm}^3

. Subtracting the volume of the central cylindrical hole of radius

4\text{ cm}

and height

15\text{ cm}

, which is

\pi(4^2)(15) = 240\pi\text{ cm}^3

, yields

1040\pi - 240\pi = 800\pi\text{ cm}^3

Adım Adım Çözüm

Calculate the total volume of the conical frustum before drilling.

V_{\text{frustum}} = \frac{1}{3}\pi h (R^2 + r^2 + R r) = \frac{1}{3}\pi (15)(12^2 + 4^2 + 12 \times 4) = 5\pi(144 + 16 + 48) = 1040\pi\text{ cm}^3

The total volume of a frustum of a cone is determined by its vertical height and the radii of its top and bottom circular bases.

Calculate the volume of the cylindrical hole drilled through the solid.

V_{\text{cylinder}} = \pi r_{\text{hole}}^2 h = \pi (4^2)(15) = 240\pi\text{ cm}^3

The drilled hole forms a right circular cylinder of radius

4\text{ cm}

and height equal to the full height of the frustum (

15\text{ cm}

Subtract the cylinder's volume from the frustum's volume to find the remaining volume.

V_{\text{remaining}} = V_{\text{frustum}} - V_{\text{cylinder}} = 1040\pi - 240\pi = 800\pi\text{ cm}^3

Removing material by drilling decreases the overall volume of the original solid by the exact volume of the cylindrical bore.

Anahtar Kavram

Volume of Composite Solids and Conical Frustums

Tahmini Süre:2m 30s

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