August 2023 Geometry Regents Part III

This exam was adminstered in August 2023.

August 2023 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

32. Josh is making a square-based fire pit out of concrete for his backyard, as modeled by the right prism below. He plans to make the outside walls of the fire pit 3.5 feet on each side with a height of 1.5 feet. The concrete walls of the fire pit are going to be 9 inches thick.

If a bag of concrete mix will fill 0.6 ft³, determine and state the minimum number of bags needed to build the fire pit.

Answer:

First you need to find the Volume, which is length * width * height (rember that it has a sqware base) minus the volume of the hole in the center, which has the same height but is 18 inches less wide and long. Also keep in mind that they are mixing feet and inches in this problem. (Guess what! Real life does that, too!)

Nine inches is .75 feet, and 18 inches is 1.5 feet. Thus, 3.5 - 1.5 = 2.0 feet.

Volume = (3.5)(3.5)(1.5) - (2)(2)(1.5) = 12.375 ft³

If one bag of concrete mix fills 0.6 ft³, then the required number will be 12.375/0.6 = 20.625 or 21 when rounded up. You must round up or you will not have enough concrete mix to complete the job.

33. A telephone pole 11 meters tall needs to be stabilized with a support beam, as modeled below.

Two conditions for proper support are:

• The beam reaches the telephone pole at 70% of the telephone pole’s height above the ground.

• The beam forms a 65° angle with the ground.

Determine and state, to the nearest tenth of a meter, the length of the support beam that meets these conditions for this telephone pole.

Determine and state, to the nearest tenth of a meter, how far the support beam must be placed from the base of the pole to meet the conditions.

Answer:

First, find 70% of 11 meters (by multiplying) to find how far up the pole the beam will reach. Then use the sin 65° to find the length of the beam, and either tan 65° or the Pythagorean Theorem to find the distance on the ground.

11 * .7 = 7.7 meters.

sin 65° = 7.7 / x
x = 7.7 / sin 65° = 8.46 = 8.5 meters

tan 65° = 7.7 / y
y = 7.7 / tan 65° = 3.59 = 3.6 meters

Look at these answers. The beam is 8.5 meters, which is reasonable for the hypotenuse when the longer leg is 7.7. And the base is 3.6 meters, which is less than half of 7.7, which is reasonable for the side opposite a 25 degree angle. If the numbers you got seem extremely large or extremely small, you may have done one of three things: used radians instead of degrees, entered the ratio incorrectly (or used the wrong one), or forgot a decimal point somewhere.

If you got a negative answer, it's probably radians.

34. The coordinates of the vertices of quadrilateral ABCD are A(0,4), B(3,8), C(8,3), and D(5,-1).

Prove that ABCD is a parallelogram, but not a rectangle.
[The use of the set of axes below is optional.]

Answer:

Use the grid if you want to visualize the problem better. You can also use it to help with finding slopes.

A parallelogram has two pairs of parallel sides, and those sides will therefore have the same slope. In a rectangle, the sides must be perpendicular, which means that the slopes are inverse reciprocals and that the products of the slopes of the two lines will be -1.

Find all four slopes. Show that the opposite sides have the same slope. Show that the consecutive sides are not perpendicular. Then write your concluding statement.

Slope of AB = (8-4) / (3-0) = 4/3. Slope of BC = (3-8) / (8-3) = -5/5 = -1.
Slope of CD = (-1-3) / (5-8) = -4/-3 = 4/3. Slope of DA = (4-(-1)) / (0-5) = 5/-5 = -1.

AB || CD and BC || DA because they have the same slopes, therefore ABCD is a parallelogram.

Slope of AB * Slope of BC = (4/3)(-1) = -4/3. The slopes are not inverse reciprocals, so the lines are not perpendicular. Therefore, ABCD is NOT a rectangle.

Anther method:

Paralelograms have a lot of properties, as do rectangles. If you don't mind use the distance formula (Pythagorean Theorem, if you graphed ABCD), you can show that the opposites sides are congruent, which makes it a parallelogram, but the diagonals are NOT congruent, which means is could not be a rectangle.

This is a valid option. However, I'd prefer to use the slope formula over the distance formula is I could. You basically have to find the slope to use the distance formula.

End of Part III

How did you do?

Questions, comments and corrections welcome.

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