Question

You work for a roofing company and must order the correct number of tiles to complete the final side of the roof. It is in the shape of a trapezoid. The numbers of tiles in each row form a sequence. We know we will have 20 rows to complete the job. The first row has ten tiles. Each row has two more tiles than the previous row.

a. Is this sequence arithmetic or geometric? Explain.
b. How many tiles are needed for each of the first four rows?
c. Write the equation for the sequence in this situation.
d. How many tiles are in the fifteenth row?
e. How many total tiles will be needed to complete the job?

Answer 1

You are visiting a Redwood tree forest and want to verify the height of one of the trees. You measure its shadow along the ground and use trig to calculate the height.

The shadow measures 500 feet and you calculate the angle of elevation to be 35 degrees, This forms a right triangle.

a. What is the measure of the other acute angle?

b. What is the height of the tree?

c. You are standing at the end of the tree's shadow and want to take a picture of the tree but your camera can only focus at a distance of fewer than 500 feet. When you hold the camera to take the picture it is 5 feet above the ground. What is the distance from the end of the shadow to the top of the tree?

d. Can you take a clear picture of the top of the tree from where you are standing?

e. How many total tiles will be needed to complete the job?

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Part 1

Yes, it is an arithmetic sequence: 10 + 2(n-1) where n is the row number.

You can use this expression to answer the other 2 parts.

Part 2

The other angle is 55 degrees and the height is 500 tan 35.

You can use this information to answer the other parts of the question.

Answer 2

a. This is an arithmetic sequence, because the difference between consecutive terms in the sequence is constant. Specifically, the difference between consecutive terms is 2 tiles.

b. The first row has 10 tiles, the second row has 12 tiles, the third row has 14 tiles, and the fourth row has 16 tiles.

c. The general formula for the nth term of an arithmetic sequence with first term a1 and common difference d is a_n = a_1 + (n-1)d. In this situation, a1 = 10 (the first row has 10 tiles) and d = 2 (each subsequent row has 2 more tiles than the previous row). Therefore, the equation for the sequence is a_n = 10 + (n-1)2.

d. The fifteenth row has 30 tiles, which can be calculated using the equation a_n = 10 + (n-1)2 with n = 15.

e. To find the total number of tiles needed to complete the job, we need to find the sum of the terms in the sequence for all the rows. The sum of the terms in an arithmetic sequence is equal to the average of the first and last term, multiplied by the number of terms. In this case, the average of the first and last term is (10 + 30)/2 = 20, and the number of terms is 20 rows. Therefore, the total number of tiles needed to complete the job is 20 * 20 = 400 tiles.