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Answer :
Sure! Let's go through the problem step-by-step.
1. Understanding the problem:
- The airplane takes off 12.5 miles south of a city.
- It flies due north at a constant speed of 170 miles per hour.
- We need to determine the time it takes for the airplane to be 115 miles north of the city.
2. Calculating the total distance:
- Since the airplane starts 12.5 miles south of the city and needs to be 115 miles north, the total distance it needs to travel is the sum of these distances: [tex]\( 12.5 + 115 = 127.5 \)[/tex] miles.
3. Setting up the equation:
- The airplane flies at a constant speed of 170 miles per hour.
- We denote the time it takes to cover the distance as [tex]\( x \)[/tex] hours.
- The total distance traveled can be represented as [tex]\( 170x \)[/tex] (since distance traveled = speed [tex]\(\times\)[/tex] time).
4. Formulating the equation:
- The distance equation we get is:
[tex]\[
170x = 127.5
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{127.5}{170}
\][/tex]
5. Matching the correct option:
- Let's look at the given options and match which one corresponds to our equation.
- Option (A): [tex]\( 170 + 12.5x = 115 \)[/tex]
- This does not match our setup.
- Option (B): [tex]\( 170 - 12.5x = 115 \)[/tex]
- This does not fit either.
- Option (C): [tex]\( 170x + 12.5 = 115 \)[/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
170x + 12.5 = 115 \quad \Rightarrow \quad 170x = 115 - 12.5 \quad \Rightarrow \quad 170x = 102.5 \quad \Rightarrow \quad x = \frac{102.5}{170}
\][/tex]
- While similar, this doesn’t match our total distance.
- Option (D): [tex]\( 170x - 12.5 = 115 \)[/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
170x - 12.5 = 115 \quad \Rightarrow \quad 170x = 115 + 12.5 \quad \Rightarrow \quad 170x = 127.5 \quad \Rightarrow \quad x = \frac{127.5}{170}
\][/tex]
- This matches perfectly with our earlier setup.
Therefore, the correct equation is:
[tex]\[
170x - 12.5 = 115
\][/tex]
So, the correct answer is:
(D) [tex]\( 170x - 12.5 = 115 \)[/tex]
1. Understanding the problem:
- The airplane takes off 12.5 miles south of a city.
- It flies due north at a constant speed of 170 miles per hour.
- We need to determine the time it takes for the airplane to be 115 miles north of the city.
2. Calculating the total distance:
- Since the airplane starts 12.5 miles south of the city and needs to be 115 miles north, the total distance it needs to travel is the sum of these distances: [tex]\( 12.5 + 115 = 127.5 \)[/tex] miles.
3. Setting up the equation:
- The airplane flies at a constant speed of 170 miles per hour.
- We denote the time it takes to cover the distance as [tex]\( x \)[/tex] hours.
- The total distance traveled can be represented as [tex]\( 170x \)[/tex] (since distance traveled = speed [tex]\(\times\)[/tex] time).
4. Formulating the equation:
- The distance equation we get is:
[tex]\[
170x = 127.5
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{127.5}{170}
\][/tex]
5. Matching the correct option:
- Let's look at the given options and match which one corresponds to our equation.
- Option (A): [tex]\( 170 + 12.5x = 115 \)[/tex]
- This does not match our setup.
- Option (B): [tex]\( 170 - 12.5x = 115 \)[/tex]
- This does not fit either.
- Option (C): [tex]\( 170x + 12.5 = 115 \)[/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
170x + 12.5 = 115 \quad \Rightarrow \quad 170x = 115 - 12.5 \quad \Rightarrow \quad 170x = 102.5 \quad \Rightarrow \quad x = \frac{102.5}{170}
\][/tex]
- While similar, this doesn’t match our total distance.
- Option (D): [tex]\( 170x - 12.5 = 115 \)[/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
170x - 12.5 = 115 \quad \Rightarrow \quad 170x = 115 + 12.5 \quad \Rightarrow \quad 170x = 127.5 \quad \Rightarrow \quad x = \frac{127.5}{170}
\][/tex]
- This matches perfectly with our earlier setup.
Therefore, the correct equation is:
[tex]\[
170x - 12.5 = 115
\][/tex]
So, the correct answer is:
(D) [tex]\( 170x - 12.5 = 115 \)[/tex]
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