Math Help - Questions on Geometry.

1. Questions on Geometry.

1.)
A student who is 5 feet tall stands 40 feet away from the base of a water tower. The angle of elevation between the top of the student’s head and the top of the water tower is 62 degrees. How tall is the water tower, to the nearest foot?

I know this has to involve trig, but I dont know which functions.

2.)
The size of a television is based on the length of the diagonal across the screen. The diagonal of Sara’s television screen is 32 inches and width is 25 inches. The border aound the screen is ½ inch wide. The television opening I her entertainment cabinet is 22 inches high and 27 inches wide. Will the television fit into her open-backed cabinet? Why?

I used the Pythagorean theorem to find the height, and I say it will fit in. The height will be 19.5. I think..

3.) Polygon ABCD is congruent to polygon PQRS, where angle A corresponds to angle P. The value of angle A is 5x-25 and the value of angle P is 3x+5. What is the measure of angle A?
System of equations?

4.) As Alice walks home she passes an empty rectangular field. She can either walk around the edge of the field or across the field diagonally. About how many feet less is the walk diagonally across the lot?
Dimensions:
Length: 400 feet
Width: 250 feet.

Im assumiung use the Pythag theorem and substract the difference.

400^2+250^2=C^2

178 feet difference?

2. For the water tower problem...

Draw two triangles. One starting from the ground and going to the boy's head. The other should contain the first triangle, but should continue to the top of the water tower. The angle between the base of the triangles and the hypotenuse should be 62.

Look at the smaller triangle first. Since the boy is 5 ft tall, the height is 5. The angle is 62. The tangent of the angle is equal to the height / base. Solve for the base of the smaller triangle.

Now you know the base of the larger triangle (base of smaller + 40 ft)
Use tangent again to solve for the height of the large triangle.

Hope this helps - if it doesn't I can draw a picture just post.

3. 1) consider the triangle with base as the 40 feet horizontal distance and the height as the height of the water tower minus the 5 feet height of the person. you are given the base and height of the right triangle, plus an acute angle, so use tan(62), and the rest follows.

2) You are correct, it will fit. But when applying the pythagorean theorem to the diagonal of 32 and the width of 25, the resulting length is sqrt(399).

3) This is a simple algebra question. It's just 5x-25 = 3x+5 and solve.

4) Yes, you got this question and concept correct, it's shorter by about 178.3009 feet. In general, make sure you add some additional significant figures after the decimal place instead of just saying 178.

4. Originally Posted by NYCKid09
1.)
A student who is 5 feet tall stands 40 feet away from the base of a water tower. The angle of elevation between the top of the student’s head and the top of the water tower is 62 degrees. How tall is the water tower, to the nearest foot?

I know this has to involve trig, but I dont know which functions.
$\tan\theta=\frac{opposite}{adjacent}$
$\theta$ is given to you and so is the adjacent side. I'll leave the rest up to you. (Remember the student is 5 feet tall!)

Originally Posted by NYCKid09
2.)
The size of a television is based on the length of the diagonal across the screen. The diagonal of Sara’s television screen is 32 inches and width is 25 inches. The border aound the screen is ½ inch wide. The television opening I her entertainment cabinet is 22 inches high and 27 inches wide. Will the television fit into her open-backed cabinet? Why?

I used the Pythagorean theorem to find the height, and I say it will fit in. The height will be 19.5. I think..
The Pythagorean theorem is a good choice! Don't forget to add the width of the border.

Originally Posted by NYCKid09
3.) Polygon ABCD is congruent to polygon PQRS, where angle A corresponds to angle P. The value of angle A is 5x-25 and the value of angle P is 3x+5. What is the measure of angle A?

50?
Precisely!
Originally Posted by NYCKid09
4.) As Alice walks home she passes an empty rectangular field. She can either walk around the edge of the field or across the field diagonally. About how many feet less is the walk diagonally across the lot?
Dimensions:
Length: 400 feet
Width: 250 feet.

Im assumiung use the Pythag theorem and substract the difference.

400^2+250^2=C^2

178 feet difference?
You are once again correct!
Oh boy! you're good! Keep it UP!

5. Originally Posted by NYCKid09
1.)
A student who is 5 feet tall stands 40 feet away from the base of a water tower. The angle of elevation between the top of the student’s head and the top of the water tower is 62 degrees. How tall is the water tower, to the nearest foot?

I know this has to involve trig, but I dont know which functions.

2.)
The size of a television is based on the length of the diagonal across the screen. The diagonal of Sara’s television screen is 32 inches and width is 25 inches. The border aound the screen is ½ inch wide. The television opening I her entertainment cabinet is 22 inches high and 27 inches wide. Will the television fit into her open-backed cabinet? Why?

I used the Pythagorean theorem to find the height, and I say it will fit in. The height will be 19.5. I think..

3.) Polygon ABCD is congruent to polygon PQRS, where angle A corresponds to angle P. The value of angle A is 5x-25 and the value of angle P is 3x+5. What is the measure of angle A?

50?

4.) As Alice walks home she passes an empty rectangular field. She can either walk around the edge of the field or across the field diagonally. About how many feet less is the walk diagonally across the lot?
Dimensions:
Length: 400 feet
Width: 250 feet.

Im assumiung use the Pythag theorem and substract the difference.

400^2+250^2=C^2

178 feet difference?
For #1:

You will need to use similar triangles for this problem. First draw a picture. Draw a person, and mark it 5 feet tall. Then draw a line from the person's feet to the base of the tower, then draw your tower and draw a line from the FEET of the person to the top of the water tower. This is your first triangle.

Next, draw another triangle on top of the one you have just drawn. This one will be another right triangle with hypotenuse from top of person's head to top of water tower, and the legs will be from the top of the person's head to the water tower (parallel to the base of the first triangle & same length 40ft) the other leg of the triangle will lay on top of the leg that makes the tower.

So, you need to find the height of the water tower. But you are not given any information about the triangle that can give you that height. So you must work with the smaller triangle at the person's head. We want to find the length of the side that is the water tower and then add 5 feet to that to make up for the height of the person. Let's call the side where the water tower is "x".

The angle of elevation is the angle above horizontal that an observer must look to see an object that is higher than he is. So, you know that your angle of elevation from the person's head to the top of the tower (at the person's head) is 62 deg. So, write that on your drawing. You also know that the base of that triangle is 40 ft.
$tan 62 = \frac{x}{40}$