AoPS Algebra & Geometry Worksheet for 17-Year-Old — Chaucer Astrolabe & Pythagorean Theorem

Instructions

Read each problem carefully. These problems integrate concepts from algebra, geometry, and even a bit of history. Show your work where appropriate to understand the solution process. Use your knowledge of algebraic manipulation, geometric properties, and the Pythagorean theorem to find the solutions.

Section 1: Advanced Algebraic Manipulation

1. If x and y are real numbers such that x + y = 6 and x² + y² = 20, what is the value of x³ + y³? (Hint: Consider the expansion of (x + y)² and the factorization of x³ + y³.)

Section 2: The Geometry-Algebra Connection

2. A rectangular prism (a box) has a length of 12 cm, a width of 5 cm, and a height of 7 cm. What is the length of the longest interior diagonal of the prism, connecting one corner to the opposite corner? Express your answer in simplest radical form.
3. The length of a rectangle is 5 inches more than its width. A square has a side length equal to the width of the rectangle. If the area of the rectangle is 36 square inches greater than the area of the square, what are the dimensions of the rectangle?

Section 3: A Historical Application

4. In his 1391 work, "A Treatise on the Astrolabe," Geoffrey Chaucer explained how to use the device to determine the height of a tall object. The method relies on the principles of right-triangle geometry.
   
   Imagine you are standing 60 feet away from the base of a tall tower. Using a simplified astrolabe, you measure the angle of elevation from your eye level to the top of the tower to be 45°. Your own eye level is 5 feet above the ground. What is the total height of the tower?
   
   (Note: For a 45° angle in a right triangle, the two legs opposite the non-right angles are equal in length.)

Answer Key

1. Solution:

We are given x + y = 6 and x² + y² = 20.
First, find the value of xy. We know that (x + y)² = x² + 2xy + y².
Substitute the given values:
(6)² = (x² + y²) + 2xy
36 = 20 + 2xy
16 = 2xy
xy = 8

Now, we need to find x³ + y³. The factorization for the sum of cubes is x³ + y³ = (x + y)(x² - xy + y²).
We can rearrange the terms to use the values we know: x³ + y³ = (x + y)((x² + y²) - xy).
Substitute the known values into this equation:
x³ + y³ = (6)(20 - 8)
x³ + y³ = (6)(12)
x³ + y³ = 72

Answer: 72

2. Solution:

This problem requires a three-dimensional application of the Pythagorean theorem. Let the length be l=12, width be w=5, and height be h=7.
First, find the diagonal of the base of the prism (let's call it d_b). This diagonal forms a right triangle with the length and width.
d_b² = l² + w²
d_b² = 12² + 5² = 144 + 25 = 169
d_b = √169 = 13 cm

Now, the interior diagonal of the prism (let's call it D) forms a new right triangle with the base diagonal (d_b) and the height (h).
D² = d_b² + h²
D² = 169 + 7² = 169 + 49 = 218
D = √218
Since 218 = 2 * 109 (and 109 is prime), the radical cannot be simplified further.
(Alternatively, the formula for a space diagonal is D = √(l² + w² + h²), which yields √(12² + 5² + 7²) = √(144 + 25 + 49) = √218.)

Answer: √218 cm

3. Solution:

Let w be the width of the rectangle.
The length of the rectangle is l = w + 5.
The area of the rectangle is A_rect = l × w = (w + 5)w = w² + 5w.

The side length of the square is also w.
The area of the square is A_sq = w².

We are told that the area of the rectangle is 36 square inches greater than the area of the square.
A_rect = A_sq + 36
w² + 5w = w² + 36
Subtract w² from both sides:
5w = 36
w = 36 / 5 = 7.2 inches

Now find the length:
l = w + 5 = 7.2 + 5 = 12.2 inches.

Answer: The rectangle has a width of 7.2 inches and a length of 12.2 inches.

4. Solution:

This problem can be modeled with a right triangle.
The distance from you to the tower is one leg of the triangle: base = 60 feet.
The height of the tower above your eye level is the other leg of the triangle: let's call it h_adj.
The angle of elevation is 45°.

In a right triangle, tan(θ) = opposite / adjacent.
tan(45°) = h_adj / 60
We know that tan(45°) = 1. (As noted in the hint, for a 45° angle, the two legs are equal).
1 = h_adj / 60
h_adj = 60 feet.

This is the height of the tower from your eye level up. To find the total height of the tower, you must add the height of your eye level from the ground.
Total Height = h_adj + eye level height
Total Height = 60 feet + 5 feet = 65 feet.

Answer: The total height of the tower is 65 feet.