1.Which angle measure is consistent with all of the following criteria?
(i) The reference angle is 40°.
(ii) The angle is more than one revolution.
(iii) The angle's terminal side is in Quadrant II.
(iv) The angle isn't positive.
A. –580°
B. –500°
C. –220°
D. 500°
2.0 The radius of a circle is 7.2 cm. If a central angle measures 112°, what is the length of its intercepted arc
to the nearest hundredth?
A. 14.07 cm
B. 7.04 cm
C. 2.24 cm
D. 4.48 cm
3. Convert 21°50" to decimal degrees. Round your answer to the nearest thousandth.
A. 21.0138°
B. 21.014°
C. 21.8333°
D. 21.833°
4. A carpenter wants to be sure that the corner of a building is square and measures 6.0 ft and 8.0 ft along
the sides. How long should the diagonal be?
A. 12 ft
B. 10 ft
C. 11 ft
D. 14 ft
5. A wheel 5.00 ft in diameter rolls up a 15.0° incline. How far above the base of the incline is the top of
the wheel after the wheel has completed one revolution?
A. 4.07 ft
B. 13.1 ft
C. 8.13 ft
D. 9.07 ft
Q. 1 c) -220 degrees. is -180 Plus(-40) degrees. It will be in Quadrant II
Q. 2 a) 14.07 cms
radius = 7.2 cm perimieter 2 x 3.14 x 7.2 = 45.216 cm
angle is 112 degree
112/360x45.216 = 14.0672 or 14.07 cm
Q.3 d) 21.833
50/60 is 0.833 (nearest thousandth)
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1. To determine which angle measure is consistent with all of the given criteria, we can break down each criterion:
(i) The reference angle is 40°: The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. Since the reference angle is 40°, the angle itself must be in Quadrant I or Quadrant IV.
(ii) The angle is more than one revolution: One revolution is equal to 360°. So, an angle greater than one revolution would be more than 360°.
(iii) The angle's terminal side is in Quadrant II: In Quadrant II, the terminal side of an angle is above the x-axis and to the left of the y-axis.
(iv) The angle isn't positive: Based on the previous criteria, the angle must be in Quadrant II (since its terminal side is in Quadrant II) and it must be greater than 360° (since it's more than one revolution). Therefore, the angle will be negative.
Putting all the criteria together, the angle measure that satisfies all the given conditions is angle A. –580°.
2. To find the length of an intercepted arc in a circle, we need to know the radius of the circle and the measure of the central angle.
In this case, the radius of the circle is given as 7.2 cm and the central angle measures 112°.
The formula to find the length of an intercepted arc is: Length of arc = (Angle measure / 360°) * (2 * π * radius)
Substituting the given values, we have: Length of arc = (112° / 360°) * (2 * π * 7.2 cm)
Calculating this expression gives us: Length of arc ≈ 14.07 cm
Therefore, the length of the intercepted arc is approximately 14.07 cm, which corresponds to option A.
3. To convert an angle measure given in degrees, minutes, and seconds (DMS) to decimal degrees, we can use the following conversion factors:
1° = 60 minutes (')
1' = 60 seconds (")
For the given angle measure of 21°50", we need to convert the minutes and seconds to decimal form and add them to the degrees.
Converting the minutes: 50" ÷ 60 = 0.8333'
Converting the seconds: There are no further conversions needed since seconds are already in decimal form.
Adding the converted minutes and seconds to the degrees: 21° + 0.8333' ≈ 21.8333°
Rounded to the nearest thousandth, the decimal degree equivalent of 21°50" is approximately 21.833°, corresponding to option D.
4. To find the length of the diagonal of a rectangle given the lengths of its sides, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) is equal to the square of the length of the longest side (hypotenuse or diagonal).
In this case, the two sides of the rectangle measure 6.0 ft and 8.0 ft. Let's label them as side a = 6.0 ft and side b = 8.0 ft.
Using the Pythagorean theorem, we can calculate the diagonal (d) as follows:
d² = a² + b²
d² = 6.0² + 8.0²
d² = 36.0 + 64.0
d² = 100.0
Taking the square root of both sides, we find:
d = √100.0
d = 10.0 ft
Therefore, the length of the diagonal should be 10 ft, which corresponds to option B.
5. To determine how far above the base of the incline the top of the wheel is after one revolution, we need to calculate the vertical distance covered by the wheel.
Given that the diameter of the wheel is 5.00 ft, we first need to find the radius of the wheel:
radius = diameter / 2 = 5.00 ft / 2 = 2.50 ft
The vertical distance covered by the wheel after one revolution can be calculated using the formula:
distance = radius * sin(angle)
In this case, the angle of the incline is given as 15.0°. We need to convert this angle to radians before using it in the formula since the trigonometric function expects angles in radians.
Angle in radians = angle in degrees * (π/180)
Angle in radians = 15.0° * (π/180) ≈ 0.2618 radians
Substituting the values into the formula, we have:
distance = 2.50 ft * sin(0.2618)
distance ≈ 2.50 ft * 0.2553
distance ≈ 0.6383 ft
Therefore, the top of the wheel is approximately 0.64 ft above the base of the incline after the wheel has completed one revolution, which corresponds to option A.