Answer:
- x = 2
- r = 48
- 3536 ft
- √10
- 10°
Step-by-step explanation:
1. The Pythagorean theorem applies, so the sum of the squares of the legs is equal to the square of the hypotenuse.
x² + (x+3)² = (√29)²
2x² +6x +9 = 29 . . . . . . eliminate parentheses, collect terms
2x² +6x -20 = 0 . . . . . . subtract 29
x² +3x -10 = 0 . . . . . . . . divide by 2
(x +5)(x -2) = 0 . . . . . . . . factor
x = -5 or +2 . . . . . . . . . . . . values that make the factors zero (-5 is extraneous)
The value of x is 2.
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2. Even though the figure is drawn incorrectly the Pythagorean theorem still applies. (The second attachment shows the circle in the right place. The figure is not to scale.)
14² + r² = (r+2)²
196 + r² = r² +4r +4 . . . . eliminate parentheses
192 = 4r . . . . . . . . . . . . . .subtract r²+4 from both sides
48 = r . . . . . . . . . . . . . . . . divide by 4
The value of r is 48.
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3. You will recognize that the triangle involving the distance above the ground, the distance along the ground, and the straight-line bullet path is an isosceles right triangle. So, the hypotenuse is √2 times the length of either leg. The path length of the bullet is ...
(2500 ft)√2 ≈ 3536 ft
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4. see below for the diagram
The length to the terminal point is found using the Pythagorean theorem:
d = √(3² +(-1)²) = √10
The distance from the origin to the terminal point is √10 ≈ 3.16.
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5. Angles "a" and "8a" total 90°, so we have ...
a + 8a = 90°
9a = 90° . . . . . . . . collect terms
a = 10° . . . . . . . . . . divide by 9
The value of a is 10°.
