Question

A chain fits tightly around two gears as shown. What is the radius of the smaller gear? Round your answer to the nearest tenth.

Answer 1

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Answer 2

Answer:

Option A is correct

4.8 in

Step-by-step explanation:

Definition:

If the centers of two circle radius r and r' are d units apart, then the length of the direct common tangent between them is:

$\sqrt{d^2-(r-r')^2}$                 ....[1]

As per the statement:

A chain fits tightly around two gears as shown.

Let r' be the radius of the smaller gear.

The radius of lager gear(r) = 14 in

The distance between the centers(d) = 22 in.

Direct common tangent = 20 in.

Substitute in [1] we have;

$20 = \sqrt{(22)^2-(14-r')^2}$

Squaring both sides we have;

$20^2 = 484-(14-r')^2$

$400= 484-(14-r')^2$

⇒$(14-r')^2 = 84$

Taking square root both sides we have;

$14-r' = \sqrt{84}$

⇒$14-r' = 9.2$

Subtract 14 from both sides we have;

$-r' = -4.8$

Divide both sides by -1 we have;

$r' = 4.8$ inches

Therefore, the radius of the smaller gear to the nearest tenth is, 4.8 inches