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
I did #1 for you (see related links below). These questions cover a variety of techniques, so you must have studied several topics. Have you no ideas at all on their workings? If not, a little help here may not be sufficient to get you through.
1. To find the area of triangle ABC, we can use the formula for the area of a triangle given two sides and the included angle:
Area = 0.5 * a * b * sin(β)
In this case, we have a = 10.2 cm, β = 57.8°, and the side b is not given. So, we need to find the side b using the Law of Sines. The Law of Sines states:
sin(a) / A = sin(b) / B = sin(c) / C
In triangle ABC, we have angle a = 47.0°, side A = 10.2 cm, and angle β = 57.8°. Let's solve for side b using the Law of Sines:
sin(a) / A = sin(β) / b
sin(47.0°) / 10.2 cm = sin(57.8°) / b
Now, we can solve for b:
b = (10.2 cm * sin(57.8°)) / sin(47.0°)
Using a calculator, we find that b is approximately 15.64 cm.
Now that we have the values of a = 10.2 cm, b = 15.64 cm, and β = 57.8°, we can substitute them into the formula for the area of the triangle:
Area = 0.5 * 10.2 cm * 15.64 cm * sin(57.8°)
Using a calculator, we find that the area is approximately 58.2 cm².
Therefore, the answer is A. 58.2 cm².
2. In triangle ABC, we have angle β = 41°, angle g = 14°, and side a = 5.0. To find side c, we can use the Law of Cosines, which states:
c² = a² + b² - 2ab * cos(β)
In this case, side b is not given. So, we need to find side b using the Law of Sines. However, we can only use the Law of Sines if we know another angle-side pair. Unfortunately, we don't have that information in this question. Therefore, we cannot determine the value of c without more information.
The answer is N/A (not applicable).
3. To resolve a vector v, given its magnitude and angle, we can use the trigonometric functions cosine and sine. If the angle is measured counterclockwise from the positive x-axis, we use the cosine function to find the x-component and the sine function to find the y-component.
Given a vector v with magnitude 2.0 × 10² and angle 60°, we can resolve it as follows:
x-component = magnitude * cos(angle)
y-component = magnitude * sin(angle)
Plugging in the values:
x-component = (2.0 × 10²) * cos(60°)
y-component = (2.0 × 10²) * sin(60°)
Using a calculator to evaluate the trigonometric functions, we find that the x-component is approximately 100 and the y-component is approximately 170.
Therefore, the resolved vector v is given by:
v = 100i + 170j
The answer is C. v = 100i + 170j.
4. To find the area of triangle ABC, we can use Heron's formula. Heron's formula states that the area of a triangle with sides a, b, and c is given by:
Area = √(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, we have a = 83.4 ft, b = 53.1 ft, and c = 37.2 ft. Let's calculate the semi-perimeter:
s = (83.4 ft + 53.1 ft + 37.2 ft) / 2
s = 173.7 ft / 2
s = 86.85 ft
Now, we can plug the values into Heron's formula:
Area = √(86.85 ft(86.85 ft - 83.4 ft)(86.85 ft - 53.1 ft)(86.85 ft - 37.2 ft))
Using a calculator to evaluate the expression, we find that the area is approximately 709 ft².
Therefore, the answer is B. 709 ft².
5. Let's go through each statement to determine if it is true or false:
(i) The magnitude of a vector is based on the Pythagorean theorem.
True. The magnitude of a vector is the square root of the sum of the squares of its components. This is the same concept as the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
(ii) The law of cosines is based on the Pythagorean theorem.
True. The law of cosines is a generalization of the Pythagorean theorem for any triangle. It relates the lengths of the sides and the cosine of one of the angles.
(iii) The law of sines is based on the Pythagorean theorem.
False. The law of sines is not based on the Pythagorean theorem. It relates the ratios of the lengths of the sides of a triangle to the sines of the opposite angles.
Therefore, the correct answer is C. (ii) only.
6. To calculate the work done in raising a box vertically, we need to use the formula:
Work = force * distance
In this case, the force is the weight of the box, and the distance is the vertical distance it is raised.
The weight of the box is given as 117 lb. To convert this to foot-pounds, we need to multiply by the acceleration due to gravity, which is approximately 32.2 ft/s². So, the weight in foot-pounds is:
117 lb * 32.2 ft/s² = 3767.4 ft-lb
The distance the box is raised is given as 18.0 inches. To convert this to feet, we divide by 12:
18.0 inches / 12 = 1.5 ft
Now, we can calculate the work done:
Work = 3767.4 ft-lb * 1.5 ft = 5641.1 ft-lb
Therefore, the answer is D. 5641.1 ft-lb.
7. In triangle ABC, we have angle g = 61.0°, angle β = 29.0°, and side b = 20.5. To find side a, we can use the Law of Sines, which states:
sin(a) / A = sin(b) / B = sin(c) / C
In this case, side A is not given. So, we need to find side A using the Law of Sines. However, we can only use the Law of Sines if we know another angle-side pair. Unfortunately, we don't have that information in this question. Therefore, we cannot determine the value of a without more information.
The answer is N/A (not applicable).