June 2024 Geometry Regents Part III

This exam was adminstered in June 2024 .

June 2024 Geometry, Part III

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

32. A building is composed of a rectangular pyramid on top of a rectangular prism, as shown in the diagram below. The rectangular prism has a length of 38 feet, a width of 15 feet, and a height of 22 feet. The rectangular pyramid sits directly on top of the rectangular prism, and its height is 12 feet.

An air purification filter was installed that will clean all the air in the building at a rate of 2400 cubic feet per minute. Determine and state how long it will take, to the nearest tenth of a minute, for the filter to clean the air contained in the building.

Answer:

First you need to find the volume of the building. The building is a pyramid sitting on top of a prism. You have to find the volumes for each of these separately and add them together.

The volume of a rectangular prism is V = (L)(W)(H), and the volume of a rectangular pyramid is V = 1/3(L)(W)(H). Note that the length and width of the two objects are the same but the height are different.

V(prism) = (38)(15)(22) = 12,540 and v(pyramid) = 1/3(38)(15)(12) = 2280. Total volume = 14820.

Next, divide this about by 2400 ft³/min. 14820/2400 = 6.175

To the nearest tenth of a minute, it will take 6.2 minutes.

33. Given: △ABC, △DEF, AB ⟂ BC, DE ⟂ EF, AE ≅ DB, and AC || FD.

Prove: △ABC ≅ △DEF

Answer:

They are expecting a two-column, or a paragraph, proof. Even as a paragraph, all the statements have to be made and the dots connected (so to speak). ou can't just BS a one- or two-sentence rationale which doesn't prove a thing.

Yes, I see a lot of those for every exam.

As always, start with the Given information. Your last statement should be the thing that you are trying to prove, along with a final reason.

Notice that there are two parallel lines, so you will likely have alternate interior angles. There are perpendicular lines, which create right angles, and all right angles are congruent. That means you only need to show that one other pair of sides are congruent to use either ASA or AAS. Since you're given AE = BD, you can pretty much figure out that you'll need to show that AB = DE.

1. △ABC, △DEF, AB ⟂ BC, DE ⟂ EF, AE ≅ DB, and AC || FD.	Given.
2. ∠ABC and ∠DEF are right angles.	Def of perpendicular lines
3. ∠ABC ≅ ∠DEF	All right angles are congruent.
4. ∠A ≅ ∠D	Alternate interior angles
5. BE ≅ BE	Reflexive property
6. AB ≅ DE	Addition postulate of congruence
7. △ABC ≅ △DEF	ASA

34. In the diagram below, a boat at point A is traveling toward the most powerful waterfall in North America, the Horseshoe Falls. The Horseshoe Falls has a vertical drop of 188 feet. The angle of elevation from point A to the top of the waterfall is 15°.

After the boat travels toward the falls, the angle of elevation at point B to the top of the waterfall is 23°. Determine and state, to the nearest foot, the distance the boat traveled from point A to point B.

Answer:

Call the base of the waterfall C. To find the length of AB, you have to find the lengths of AC and BC and then subtract AC - BC. You can find both of those lengths using tangent. (Make sure your calculator is in DEGREE mode, or you will get crazy answers.)

Tan 18 = 188 / x

x = 188 / tan 18 = 701.625...

Tan 23 = 188 / y

y = 188 / tan 23 = 442.900...

701.625 - 442.900 = 258.725, or about 259 feet to the nearest foot.

End of Part III

How did you do?

Questions, comments and corrections welcome.

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