A park ranger would like to clear more than 10 mi of trail. So far, he has cleared 2 mi. From now on, he can clear 34 mi of trail each hour.
Which statement describes the solution to this inequality?
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The ranger will need to clear the trail for more than 8 additional hours
The ranger will need to clear the trail for more than 10 2/3 additional hours
The ranger will need to clear the trail for more than 16 additional hours
The ranger will need to clear the trail for more than 13 1/3 additional hours
The ranger will need to clear the trail for more than 13 1/3 additional hours
The ranger has already cleared 2 mi of trail, so he needs to clear more than 10 - 2 = 8 additional miles of trail.
Since the ranger can clear 34 mi of trail each hour, he will need to clear the trail for more than 8 / 34 = 8/34 = 4/17 hours, which is approximately 0.235 hours.
Converting 0.235 hours to minutes, we have 60 * 0.235 = 14.1 minutes.
Therefore, the ranger will need to clear the trail for more than 14.1 additional minutes, which can be rounded up to the nearest hour.
The correct statement is:
The ranger will need to clear the trail for more than 15 additional minutes.
To find the solution to this inequality, we can set up an equation.
Let x represent the number of additional hours the ranger needs to clear the trail.
The ranger has already cleared 2 miles, and he can clear 34 miles per hour.
So, the total distance cleared after x hours can be represented as:
2 + 34x
The problem states that the ranger wants to clear more than 10 miles of trail. In equation form, this would be:
2 + 34x > 10
Now, let's solve this inequality.
Subtracting 2 from both sides, we have:
34x > 8
Next, divide both sides by 34:
x > 8/34
Simplifying this, we find:
x > 4/17
However, we need to convert this into mixed number format.
To convert 4/17 to a mixed number, divide the numerator (4) by the denominator (17). The quotient is the whole-number part of the mixed number, and the remainder becomes the numerator of the fractional part.
4 รท 17 = 0 remainder 4
So, 4/17 can be written as 0 4/17.
Therefore, the ranger will need to clear the trail for more than 0 4/17 additional hours.
Now, let's convert this mixed number back into an improper fraction.
0 4/17 = (17 * 0 + 4) / 17 = 4/17
Therefore, the solution to the inequality is that the ranger will need to clear the trail for more than 4/17 additional hours.
However, none of the given answer choices match this exact fraction.
To approximate the answer in the answer choices, we can convert 4/17 to a decimal.
Dividing 4 by 17 gives approximately 0.235.
Now, let's check which answer choice is greater than 0.235:
- The answer choice "The ranger will need to clear the trail for more than 8 additional hours" is not greater than 0.235.
- The answer choice "The ranger will need to clear the trail for more than 10 2/3 additional hours" is not greater than 0.235.
- The answer choice "The ranger will need to clear the trail for more than 16 additional hours" is greater than 0.235.
- The answer choice "The ranger will need to clear the trail for more than 13 1/3 additional hours" is greater than 0.235.
Therefore, the statement that describes the solution to this inequality is "The ranger will need to clear the trail for more than 13 1/3 additional hours."