>> A steep mountain is inclined 72 degrees to the horizontal and rises to a height
of 3,500 feet above the surrounding plane.
So using this first sentence, we can model the mountain using this right triangle here
or half of the mountain where the angle of inclination is 72 degrees labeled here.
And the height of the mountain would be 3,500 feet.
A cable car is to be installed running to the top of the mountain from a point 800 feet
out in the plane from the base of the mountain.
So the length of side AD here would be 800 feet.
So point D is where the cable car would start.
And from here it runs to the top of the mountain at point B.
So side BD represents the path of the cable car.
And our goal is to find the shortest length of the cable needed, or the length of side BD.
So let's go ahead and label side BD side C. Now looking at our diagram,
notice how we have this right triangle, triangle ABC,
and we also have this obtuse triangle, triangle ABD.
So it'll take us two steps to find the length of side C. We'll eventually apply the law
of cosines using this obtuse triangle.
But before we do this, we need to gather some additional information.
First notice how the angle here that measures 72 degrees,
and angle BAD would be supplementary or form a straight angle.
And therefore their sum must be 180 degrees.
Which means angle BAD would be equal to 180 degrees minus 72 degrees, or 180 degrees.
So this angle here is 108 degrees.
But now before we can apply the law of cosines to this obtuse triangle to find the length
of side C, since we'll be using this angle that measures 108 degrees, we have to know the length
of the two sides that form this angle.
We know side AD is 800 feet.
But we don't know the length of side AB.
Let's label side AB side M, and we can find the length of M using the right triangle,
triangle ABC in right triangle trigonometry.
Using the angle that measures 72 degrees, notice how we know the measure
of the opposite side is 3,500 feet, and side M would be the hypotenuse.
So because we're using the opposite side in the hypotenuse, we can use the sine function
to determine the length of side M. The sine of 72 degrees must be equal to the ratio
of the opposite side of the hypotenuse, or in this case 3,500 divided
by M. Remember we can do this because we are working with a right triangle.
And now to solve for M we would multiply both sides by M,
which would give us M times sine 72 degrees equals 3,500.
Divide both sides by sine 72 degrees.
And now we'll go to the calculator to determine this quotient.
We should first verify we are in degree mode, so we'll press the mode key.
Notice how degree is highlighted.
So we'll go back to the home screen and enter 3,500 divided by sine 72 degrees.
So M is approximately 3,680.12 feet.
Let's go ahead and label this in our diagram.
And now using the obtuse triangle, or this orange triangle here, we can apply the law
of cosines to determine the length of side C. Let's hit this up on the next slide.
The law of cosines is given here below.
The main thing to remember here is that if we use side A here on the left, the relationship is
that angle A would be opposite side A. And the sides that form angle A would be sides B
and C. So if we want to find the length of BD, which we called side C, we would use the law
of cosines in the form C squared equals the two other sides would be AD and side AB.
So we'd have the length of AD squared plus the length of AB squared minus two times the length
of AD times the length of AB times cosine
of the angle opposite side C, which measures 108 degrees.
Now we'll fill in the known values.
We'd have C squared equals AD square would be 800 squared.
Plus AB squared is 3,680.12 squared minus two times 800 times 3,680.12 times cosine
108 degrees.
And now we'll go back to the calculator to determine C squared.
We already know we're in degree mode.
So we'll press 800 squared plus 3,680.12 squared minus two times 800 times 3,680.12 times cosine
108 degrees.
And we already know we're in degree mode.
So we'll press enter.
So C squared is equal to 16,002,834.61.
And now we know C is the length.
So C has to be positive.
So we'll take the principle square root of both sides of the equation.
And now we'll go back to the calculator.
Remember taking the square root is the same as raising something to the one-half power.
So because we have this value already in the calculator,
we could just press the exponent key of the caret.
And then raise the previous answer to the one-half power.
And enter.
Notice how this would be equivalent to taking the square root of 16,002,835.61.
Either way, C is approximately 4,000.35 feet.
Which is the shortest length of the cable needed for the cable car.
Which again is the length of side C. So approximately 4,000.35 feet.
I hope you found this helpful. ------------------------------cd57caa99280--