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
To find the answers to these questions, we'll need to use various mathematical formulas and concepts. Here's how you can approach each question:
1. To find the area of a triangle given side lengths and an angle, we can use the formula:
Area = (1/2) * a * b * sin(C)
In this case, a = 10.2 cm, β = 57.8° (angle opposite side b), and a = 47.0° (angle opposite side a).
Plugging in the values, we get:
Area = (1/2) * 10.2 cm * 10.2 cm * sin(47.0°)
Use a scientific calculator to calculate sin(47.0°) and solve the equation to get the area in cm^2. The correct option will be the one closest to your result.
2. To find the value of side c in a triangle given angles and side lengths, we can use the Law of Sines. The formula is:
a/sin(A) = c/sin(C)
In this case, β = 41°, g = 14°, and a = 5.0.
Rewrite the equation as:
c = (a * sin(C)) / sin(A)
Plug in the values and calculate c. Round your answer to the nearest tenth. The correct option will be the one closest to your result.
3. To resolve a vector given magnitude and angle, we can use trigonometry. The vector components (v_x and v_y) can be calculated as:
v_x = v * cos(angle)
v_y = v * sin(angle)
In this case, the magnitude is 2.0 × 10^2 and the angle is 60°.
Plug in the values and simplify the equation. The answer should be in the form of i and j components. For example, "v = 120i + 160j". Compare to the given options and choose the correct one.
4. To find the area of a triangle given side lengths, we can use Heron's formula:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter of the triangle, given by:
s = (a + b + c) / 2
In this case, a = 83.4 ft, b = 53.1 ft, and c = 37.2 ft.
Calculate the semi-perimeter, substitute the values into the formula, and calculate the area. The correct answer will be the one closest to your result.
5. Determine which statements are true based on the Pythagorean theorem.
(i) The magnitude of a vector is based on the Pythagorean theorem. (True)
(ii) The law of cosines is based on the Pythagorean theorem. (True)
(iii) The law of sines is based on the Pythagorean theorem. (False)
Choose the correct option that includes the true statements.
6. To find the work done, we can use the formula:
Work = force * distance * cos(angle)
In this case, the box weighs 117 lb, and it is raised vertically 18.0 inches.
Convert the distance to feet, and plug in the values into the formula. Calculate the work done in ft-lb. The correct option will be the one closest to your result.
7. To find the value of side a in a triangle given angles and side lengths, we can use the Law of Sines. The formula is the same as in question 2:
a/sin(A) = c/sin(C)
In this case, g = 61.0°, β = 29.0°, and b = 20.5.
Rewrite the equation as:
a = (b * sin(A)) / sin(B)
Plug in the values and calculate a. Round your answer to the nearest tenth. The correct option will be the one closest to your result.
8. To find the value of side a in a triangle given angles and side lengths, we can use the Law of Sines. The formula is the same as in questions 2 and 7:
a/sin(A) = c/sin(C)
In this case, β = 105°, b = 14.0, and c = 11.0.
Rewrite the equation as:
a = (c * sin(A)) / sin(C)
Plug in the values and calculate a. Round your answer to the nearest tenth. The correct option will be the one closest to your result.
By following these steps, you should be able to find the correct answers to each question.
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