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.

4.8 in.

23.2 in.

14.0 in.

9.2 in.

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