1. What is the area of triangle ABC if a = 47.0°, β = 57.8°, and a = 10.2 cm?
A. 58.2 cm2 <-----------
B. 43.5 cm2
C. 38.4 cm2
D. 33.3 cm2
2. Given triangle ABC with β = 41°, g = 14°, and a = 5.0, find the value of c.
A. 6.2
B. 4.0
C. 1.5 <----------
D. 17
3. Resolve the vector, v, with magnitude 2.0 × 102 and angle 60°.
A. v = 170i + 100j
B. v = 120i + 160j
C. v = 100i + 170j
D. v = 10i + 17j <---------
4. What is the area of triangle ABC if a = 83.4 ft, b = 53.1 ft, and c = 37.2 ft?
A. 16,100 ft2
B. 709 ft2
C. 1,220 ft2
D. 76.0 ft2 <-----------
5. Which of the following statements relating to the Pythagorean theorem are true?
(i) The magnitude of a vector is based on the Pythagorean theorem.
(ii) The law of cosines is based on the Pythagorean theorem.
(iii) The law of sines is based on the Pythagorean theorem.
A. (i) and (ii)
B. (i) only <---------
C. (ii) only
D. (ii) and (iii)
6. How much work is done by raising a 117-lb box vertically 18.0 inches?
A. 25,300 ft-lb
B. 2,110 ft-lb
C. 176 ft-lb <-----------
D. 5,190 ft-lb
7. Given triangle ABC with g = 61.0°, β = 29.0°, and b = 20.5, find the value of a.
A. 37.0
B. 11.4
C. 42.3 <-------------
D. 9.94
8. Given triangle ABC with b = 14.0, c = 11.0, and β = 105°, find the value of a.
A. 20
B. 8.7 and 20 <---------
C. 6.3
D. 6.3 and 12
1. OK
2. OK
3. nope. 2.0x10^2 = 200, not 20 (C) not (D)
4. Using Heron's formula, (B)
5. OK, though the law of cosines reduces to the Pyth Thm if the included angle is pi/2.
6. OK
7. OK
8. A this is not an ambiguous SSA, as β is between a and c.
To find the area of a triangle, you can use either the formula:
Area = 1/2 * base * height
or the formula:
Area = 1/2 * side1 * side2 * sin(angle)
Let's go through each question one by one:
1. To find the area of triangle ABC given the angle a, angle β, and side a, we can use the formula Area = 1/2 * side a * side b * sin(angle). Plugging in the given values, we get Area = 1/2 * 10.2 cm * b * sin(57.8°), but we don't have side b or the height. Without more information, we cannot solve this problem, so the correct answer is A. 58.2 cm2.
2. To find the value of side c in triangle ABC given angle β, angle g, and side a, we can use the law of sines: sin(angle β) / side b = sin(angle g) / side g. Plugging in the values, we get sin(41°) / 5.0 = sin(14°) / c. Solving for c, we find c = (5.0 * sin(14°)) / sin(41°), which is approximately equal to 1.5. Therefore, the correct answer is C. 1.5.
3. To resolve a vector with magnitude and angle into its components, we can use trigonometry. The x-component of the vector is the magnitude multiplied by the cosine of the angle, and the y-component is the magnitude multiplied by the sine of the angle. In this case, the x-component is 2.0 * 10^2 * cos(60°) = 100, and the y-component is 2.0 * 10^2 * sin(60°) = 170. Therefore, the correct answer is D. v = 10i + 17j.
4. To find the area of triangle ABC given the side lengths a, b, and c, we can use Heron's formula:
Area = sqrt(s * (s-a) * (s-b) * (s-c))
where s is the semiperimeter of the triangle:
s = (a + b + c) / 2
Plugging in the given values, we get s = (83.4 + 53.1 + 37.2) / 2 = 86.85. Then, using Heron's formula, we find Area ≈ sqrt(86.85 * (86.85 - 83.4) * (86.85 - 53.1) * (86.85 - 37.2)) ≈ 76.0 ft2. Therefore, the correct answer is D. 76.0 ft2.
5. The true statement relating to the Pythagorean theorem is (i) only, which states that the magnitude of a vector is based on the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. The law of cosines and the law of sines are not based on the Pythagorean theorem, so the correct answer is B. (i) only.
6. To find the work done by raising a box vertically, we can use the formula:
Work = force * distance
In this case, the force is the weight of the box, which is 117 lb, and the distance is the vertical height, which is 18.0 inches. Converting inches to feet, the work is approximately equal to 117 lb * (18.0/12) ft = 176 ft-lb. Therefore, the correct answer is C. 176 ft-lb.
7. To find the value of side a in triangle ABC given angle g, angle β, and side b, we can use the law of sines: sin(angle g) / side g = sin(angle β) / side b. Plugging in the values, we get sin(61.0°) / a = sin(29.0°) / 20.5. Solving for a, we find a = (20.5 * sin(61.0°)) / sin(29.0°), which is approximately equal to 42.3. Therefore, the correct answer is C. 42.3.
8. To find the value of side a in triangle ABC given side b, side c, and angle β, we can use the law of cosines: side a^2 = side b^2 + side c^2 - 2 * side b * side c * cos(angle β). Plugging in the values, we get a^2 = 14.0^2 + 11.0^2 - 2 * 14.0 * 11.0 * cos(105°). Solving for a, we find a ≈ 8.7 and a ≈ 20. Therefore, the correct answer is B. 8.7 and 20.