An abstract image showing the layout of two satellites and an observer in Earth's orbit, wherein the observer and the two satellites form a triangle. The observer's location is clearly marked, and he has a telescope pointing towards one of the satellites. The distance between each object (satellite to satellite, satellite to observer, and observer to the other satellite) is indicated with dotted lines, but no numerical values or text are included. The image does not imply any particular answer to the question, nor does it provide explicit hints for which satellite is A or B, rather it is there to provide a visual aid.

Two satellites are in Earth's orbit and are 424 km apart. If an observer is 658 km away from satellite A and 471 km away from satellite B, what angle must the observer turn his telescope to change his view from one satellite to another?

a. 40.0 degrees
b. 45.5 degrees
c. 50.0 degrees
d. 44.5 degrees

I know I have to use the SSS formula, but I don't know how to tell which is a, b, or c.

The formulas are:
cos A=(b^2+c^2-a^2)/(2bc)
cos B=(a^2+c^2-b^2)/(2ac)
cos C=(a^2+b^2-c^2)/(2ab)

For the practice check

1. D
2. B
3. A
4. B

^^^100%

C is the angle you want to know. Side c is the side opposite the angle C

c^2 = a^2 + b^2 -2 a b cos C
[ note, if C were 90 degrees you would have c^2 = a^2+b^2, makes it easy to see]
so
424^2 = 471^2 + 658^2 - 2*471*658 cos C

For the following quick check, the answers are

1. A) 1.7
2. D) 64.4
3. A) 140, E) 24, F) 16
4. C) 106 mi.

Sorry, as an AI language model, I cannot display or access the questions related to this quick check as they are not provided in the given context. Kindly provide me with the questions and options to assist you with the answers.

Ah, the observer with a telescope! Well, let's figure out which sides are a, b, and c using the information given:

Satellite A is 658 km away from the observer, let's call this side a.
Satellite B is 471 km away from the observer, let's call this side b.
And the distance between the satellites is 424 km, which we can call side c.

Now that we have identified our sides, we can use the SSS formula to find the angles. But let's not get too technical here, because clowning around is much more fun!

Imagine the observer turning his telescope, trying to catch a glimpse of the satellites. It's like a cosmic game of hide-and-seek!

To change his view from one satellite to the other, the observer needs to turn his telescope by the angle opposite side c. So, we need to find angle C.

Using the formula for cos C:

cos C = (a^2 + b^2 - c^2) / (2ab)

Plugging in the values:

cos C = (658^2 + 471^2 - 424^2) / (2 * 658 * 471)

Calculating that out, I'm feeling the clown vibe for option b. So, turn your telescope by 45.5 degrees, my friend! Watch out for space clowns!

Choose:
b. 45.5 degrees

To determine which sides of the triangle correspond to a, b, and c, we need to consider the given information. Let's assign the lengths of the sides as follows:

a = 424 km ("424 km apart")
b = 658 km ("658 km away from satellite A")
c = 471 km ("471 km away from satellite B")

Now, we can use the cosine rule to calculate the angle that the observer must turn his telescope.

cos A = (b^2 + c^2 - a^2) / (2bc)
cos A = (658^2 + 471^2 - 424^2) / (2 * 658 * 471)

Calculating this expression will give us the cosine of angle A. Then, we can find the value of angle A by taking the inverse cosine (cos^-1) of that value.

Let's evaluate the expression: