4. For △XYZ, what is the measure of the smallest angle to the nearest tenth of a degree?
(Image: A picture of an inverted right triangle XYZ is shown. Side XY has a length of 3.5 and Side YZ has a length of 8. Side XZ is unlabeled.) (1 point)
11.5°
23.6°
90°
35.7°
The triangle is inverted, meaning the right angle is at vertex Y. Using the inverse tangent function on the ratio of the opposite side to the adjacent side, we can find the measure of the smallest angle: tan⁻¹(3.5/8) ≈ 23.6°. Answer: 23.6°.
5. Suppose that you are watching the tree warden trim branches from a large tree in your yard. He has climbed up 15 meters and his assistant is holding a rope that will be used to guide the branch when it falls. To the nearest meter, how long is the rope?
(Image: An illustration shows a ladder placed against a tree at a height of 15m to form a right triangle with the tree.)
18
26
10
40
The rope, the ladder, and the tree form a right triangle. The length of the rope is the hypotenuse of the triangle. Therefore, we can use the Pythagorean theorem to find the length of the rope:
c² = a² + b², where c is the length of the rope and a is the height of the ladder (15m). We do not know b, the distance from the base of the tree to the ladder.
However, we can assume that the ladder is leaning against the tree at a 45-degree angle (which is often the standard for leaning a ladder against a tree). This would mean that the distance from the base of the tree to the ladder is the same as the height of the ladder: b = a = 15m.
Substituting these values into the Pythagorean theorem, we get:
c² = 15² + 15²
c² = 450
c ≈ 21.2
Rounding to the nearest meter, the length of the rope is 21 meters. Answer: 21.
6. Solve. (Image: start root 2x plus 3 end root equals start root 6x minus 1 end root) (1 point)
1
2
3
(Image: Start Root 6 End Root)
To solve the equation, we need to isolate x on one side of the equation.
Starting with the given equation:
√(2x+3) = √(6x-1)
Square both sides of the equation:
2x+3 = 6x-1
Simplify by subtracting 2x and adding 1 to both sides:
4 = 4x
Divide both sides by 4:
x = 1
Therefore, the solution to the equation is x = 1. Answer: 1.
7. Solve. (Image: start root b superscript 4 baseline over 25 end root equals 4) (1 point)
2
(Image: plus-or-minus 5 Start Root 2 End Root)
3
(Image: plus-or-minus 2 Start Root 5 End Root)
Starting with the given equation:
√(b⁴/25) = 4
Simplify by squaring both sides of the equation:
b⁴/25 = 16
Multiply both sides by 25:
b⁴ = 400
Take the fourth root of both sides:
b = ±√(400)
b = ±20
Therefore, the solutions to the equation are b = 20 and b = -20. Answer: plus-or-minus 20.
8. Which graph matches the function (Image: y equals = start root 2x plus 6 end root)? (1 point)
(Image: A curve is graphed on a coordinate grid is shown. The curve starts at the point left-parenthesis 7 comma 0 right-parenthesis, moves up and to the right while curving slightly downward, and has an arrow on the end.)
(Image: A curve is graphed on a coordinate grid is shown. The curve starts at the point left-parenthesis negative 3 comma 0 right-parenthesis, moves up and to the right while curving slightly downward, and has an arrow on the end.)
(Image: A curve is graphed on a coordinate grid is shown. The curve starts at the point left-parenthesis 4 comma 0 right-parenthesis, moves up and to the right while curving slightly downward, and has an arrow on the end.)
(Image: A curve is graphed on a coordinate grid is shown. The curve starts at the point left-parenthesis 0 comma 1 right-parenthesis, moves up and to the right while curving slightly downward, and has an arrow on the end.)
The function y = √(2x+6) is a square root function. It starts at the point (−3,0), moves to the right and up as x increases, and has a vertical asymptote at x = −3.
The only graph that matches this description is the second one, which starts at the point (−3,0), moves up and to the right while curving slightly downward, and has an arrow on the end.
Therefore, the answer is: The second graph matches the function y = √(2x+6).
9. A ranger spots a forest fire while on a 45-meter observation tower. The angle of depression from the tower to the fire is 12°. To the nearest meter, how far is the fire from the base of the tower? (1 point)
212 meters
10 meters
216 meters
71 meters