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Sagot :
To solve the problem, we need to analyze the given statements and compare them with the correct mathematical results deduced from the clock face with a radius of 10 inches.
1. The central angle formed when one hand points at 1 and the other hand points at 3 is [tex]$30^{\circ}$[/tex].
This statement is incorrect. The central angle between 1 and 3 on a clock, which represents 2 hours apart, can be calculated as:
[tex]\[ \text{Central angle} = (3 - 1) \times 30^{\circ} = 2 \times 30^{\circ} = 60^{\circ} \][/tex]
Therefore, the correct central angle is 60 degrees, not 30 degrees.
2. The circumference of the clock is approximately 62.8 inches.
This statement is correct. The circumference of a clock with a radius of 10 inches can be calculated using the formula for the circumference of a circle:
[tex]\[ \text{Circumference} = 2 \pi \times \text{radius} = 2 \pi \times 10 \approx 62.8319 \, \text{inches} \][/tex]
Hence, the circumference is approximately 62.8 inches.
3. The minor arc measure when one hand points at 12 and the other hand points at 4 is [tex]$120^{\circ}$[/tex].
This statement is correct. To find the minor arc measure between 12 and 4, we count the number of hours between them and calculate the angle:
[tex]\[ 4 - 12 = -8 \quad (\text{corrected as the minor arc, considering it as a positive angle}) \quad = 4 \times 30^\circ = 120^\circ \][/tex]
Therefore, the minor arc measure between 12 and 4 is indeed 120 degrees.
4. The length of the major arc between 3 and 10 is approximately 31.4 inches.
This statement is incorrect. The length of the major arc between 3 and 10 can be determined by first calculating the central angle and then the arc length:
[tex]\[ \text{Number of hours} = 12 - (10 - 3) = 5 \quad (\text{for the major arc}) \][/tex]
[tex]\[ \text{Central angle} = 5 \times 30^\circ = 150^\circ \][/tex]
[tex]\[ \text{Length of major arc} = \text{circumference} \times \frac{7}{12} = 62.8319 \times \frac{7}{12} \approx 36.652 \, \text{inches} \][/tex]
The correct major arc length is approximately 36.7 inches, not 31.4 inches.
5. The length of the minor arc between 6 and 7 is approximately 5.2 inches.
This statement is correct. The length of the minor arc between 6 and 7 is calculated as:
[tex]\[ \text{Number of hours} = 1 \][/tex]
[tex]\[ \text{Central angle} = 1 \times 30^\circ = 30^\circ \][/tex]
[tex]\[ \text{Length of minor arc} = \text{circumference} \times \frac{1}{12} = 62.8319 \times \frac{1}{12} \approx 5.235 \, \text{inches} \][/tex]
Therefore, the minor arc length is approximately 5.2 inches.
To summarize, the three correct statements are:
1. The circumference of the clock is approximately 62.8 inches.
2. The minor arc measure when one hand points at 12 and the other hand points at 4 is 120 degrees.
3. The length of the minor arc between 6 and 7 is approximately 5.2 inches.
1. The central angle formed when one hand points at 1 and the other hand points at 3 is [tex]$30^{\circ}$[/tex].
This statement is incorrect. The central angle between 1 and 3 on a clock, which represents 2 hours apart, can be calculated as:
[tex]\[ \text{Central angle} = (3 - 1) \times 30^{\circ} = 2 \times 30^{\circ} = 60^{\circ} \][/tex]
Therefore, the correct central angle is 60 degrees, not 30 degrees.
2. The circumference of the clock is approximately 62.8 inches.
This statement is correct. The circumference of a clock with a radius of 10 inches can be calculated using the formula for the circumference of a circle:
[tex]\[ \text{Circumference} = 2 \pi \times \text{radius} = 2 \pi \times 10 \approx 62.8319 \, \text{inches} \][/tex]
Hence, the circumference is approximately 62.8 inches.
3. The minor arc measure when one hand points at 12 and the other hand points at 4 is [tex]$120^{\circ}$[/tex].
This statement is correct. To find the minor arc measure between 12 and 4, we count the number of hours between them and calculate the angle:
[tex]\[ 4 - 12 = -8 \quad (\text{corrected as the minor arc, considering it as a positive angle}) \quad = 4 \times 30^\circ = 120^\circ \][/tex]
Therefore, the minor arc measure between 12 and 4 is indeed 120 degrees.
4. The length of the major arc between 3 and 10 is approximately 31.4 inches.
This statement is incorrect. The length of the major arc between 3 and 10 can be determined by first calculating the central angle and then the arc length:
[tex]\[ \text{Number of hours} = 12 - (10 - 3) = 5 \quad (\text{for the major arc}) \][/tex]
[tex]\[ \text{Central angle} = 5 \times 30^\circ = 150^\circ \][/tex]
[tex]\[ \text{Length of major arc} = \text{circumference} \times \frac{7}{12} = 62.8319 \times \frac{7}{12} \approx 36.652 \, \text{inches} \][/tex]
The correct major arc length is approximately 36.7 inches, not 31.4 inches.
5. The length of the minor arc between 6 and 7 is approximately 5.2 inches.
This statement is correct. The length of the minor arc between 6 and 7 is calculated as:
[tex]\[ \text{Number of hours} = 1 \][/tex]
[tex]\[ \text{Central angle} = 1 \times 30^\circ = 30^\circ \][/tex]
[tex]\[ \text{Length of minor arc} = \text{circumference} \times \frac{1}{12} = 62.8319 \times \frac{1}{12} \approx 5.235 \, \text{inches} \][/tex]
Therefore, the minor arc length is approximately 5.2 inches.
To summarize, the three correct statements are:
1. The circumference of the clock is approximately 62.8 inches.
2. The minor arc measure when one hand points at 12 and the other hand points at 4 is 120 degrees.
3. The length of the minor arc between 6 and 7 is approximately 5.2 inches.
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