Math

QuestionFind the total length of two semicircles with centers RR and SS, where RS=12RS=12. Options are: (A) 8π8 \pi, (B) 9π9 \pi, (C) 12π12 \pi, (D) 15π15 \pi, (E) 16π16 \pi.

Studdy Solution

STEP 1

Assumptions1. The centers of the two semicircles are points R and. . The distance between R and is12 units.
3. The darkened curve is the combined circumference of the two semicircles.
4. The semicircles are identical and each is a half circle.

STEP 2

First, we need to find the radius of each semicircle. Since the distance between the centers of the two semicircles is equal to the diameter of one semicircle, we can find the radius by dividing this distance by2.
Radius=RS/2Radius = RS /2

STEP 3

Now, plug in the given value for RS to calculate the radius.
Radius=12/2Radius =12 /2

STEP 4

Calculate the radius.
Radius=12/2=6Radius =12 /2 =6

STEP 5

Now that we have the radius, we can find the circumference of one full circle (which is equivalent to the combined circumference of two semicircles) using the formula for the circumference of a circle, which is 2πr2\pi r.
Circumference=2πRadiusCircumference =2\pi Radius

STEP 6

Plug in the value for the radius to calculate the circumference.
Circumference=2π6Circumference =2\pi6

STEP 7

Calculate the circumference.
Circumference=2π6=12πCircumference =2\pi6 =12\piThe total length of the darkened curve, which is the combined circumference of the two semicircles, is 12π12\pi units. So, the correct option is (C) 12π12 \pi.

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