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
using c as the base, the altitude is a sinβ
since we have α and β, γ = 75.2°
c/sinγ = a/sinα, so
c = 1.2/sin47.0° sin75.2° = 1.59
area = 1/2 base*height = 1/2 * 1.59 * 10.2 * sin57.8° = 6.86 cm^2
how far do you get on the others?
Oops I see I dropped a 0. Picking up in the middle,
c = 10.2/sin47.0° sin75.2° = 13.48
area = 1/2 base*height = 1/2 * 13.48 * 10.2 * sin57.8° = 58.17 cm^2
can you please do them all? pleaseeeeeee
1. To find the area of triangle ABC, you can use the formula:
Area = (1/2) * a * b * sin(β)
where:
a = 10.2 cm
β = 57.8°
Plugging in the values into the formula, we get:
Area = (1/2) * 10.2 cm * 10.2 cm * sin(57.8°)
Area ≈ 43.5 cm^2
Therefore, the answer is B. 43.5 cm^2.
2. To find the value of c in triangle ABC, you can use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(β)
where:
a = 5.0
β = 41°
Plugging in the values into the formula, we get:
c^2 = 5.0^2 + b^2 - 2 * 5.0 * b * cos(41°)
We don't have the value of b given in the question, so we cannot calculate c. The answer is not provided in the options (A, B, C, or D).
3. To resolve a vector, v, with magnitude and angle, you can use the trigonometric functions cosine (cos) and sine (sin). The components of the vector can be calculated using the following formulas:
v_x = v * cos(θ)
v_y = v * sin(θ)
where:
v = 2.0 × 102
θ = 60°
Plugging in the values into the formulas, we get:
v_x = (2.0 × 102) * cos(60°)
v_x ≈ 100
v_y = (2.0 × 102) * sin(60°)
v_y ≈ 173.2
Therefore, the answer is C. v = 100i + 170j.
4. To find the area of triangle ABC, you can use Heron's formula:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
where:
a = 83.4 ft
b = 53.1 ft
c = 37.2 ft
First, calculate the semiperimeter, s:
s = (a + b + c) / 2
s = (83.4 ft + 53.1 ft + 37.2 ft) / 2
s ≈ 86.85 ft
Now, plug in the values into the formula:
Area = sqrt(86.85 ft * (86.85 ft - 83.4 ft) * (86.85 ft - 53.1 ft) * (86.85 ft - 37.2 ft))
Area ≈ 709 ft^2
Therefore, the answer is B. 709 ft^2.
5. (i) The statement "The magnitude of a vector is based on the Pythagorean theorem" is true. The magnitude of a vector can be found using the Pythagorean theorem by calculating the square root of the sum of the squares of its components.
(ii) The statement "The law of cosines is based on the Pythagorean theorem" is true. The law of cosines is a generalization of the Pythagorean theorem for any triangle, not just right triangles.
(iii) The statement "The law of sines is based on the Pythagorean theorem" is false. The law of sines relates the lengths of the sides of a triangle to the sines of its angles, but it does not involve the Pythagorean theorem.
Therefore, the answer is C. (ii) only.
6. To calculate the work done, you can use the formula:
Work = Force * Distance * cos(θ)
where:
Force = 117 lb (pounds)
Distance = 18.0 inches
First, convert the distance from inches to feet:
Distance = 18.0 inches * (1 ft / 12 inches)
Distance = 1.5 ft
Now, plug in the values into the formula:
Work = 117 lb * 1.5 ft * cos(0°)
Work ≈ 176 ft-lb
Therefore, the answer is C. 176 ft-lb.
7. To find the value of a in triangle ABC, you can use the Law of Sines:
a / sin(A) = b / sin(B)
where:
b = 20.5
β = 29.0°
First, find the value of the angle A:
A = 180° - β - g
A = 180° - 29.0° - 61.0°
A ≈ 90.0°
Now, plug in the values into the formula:
a / sin(90.0°) = 20.5 / sin(29.0°)
Since sin(90.0°) = 1, the equation simplifies to:
a = 20.5 / sin(29.0°)
Using a calculator, we can find:
a ≈ 42.3
Therefore, the answer is C. 42.3.
8. To find the value of a in triangle ABC, you can use the Law of Cosines:
a^2 = b^2 + c^2 - 2bc * cos(β)
where:
b = 14.0
c = 11.0
β = 105°
Plugging in the values into the formula, we get:
a^2 = 14.0^2 + 11.0^2 - 2 * 14.0 * 11.0 * cos(105°)
Solving for a, we have:
a ≈ 8.7 and 20
Therefore, the answer is B. 8.7 and 20.