A forest ranger travels up a river by boat to collect soil samples. The boat travels at a speed of v miles per hour for a distance of 2 miles. After collecting the samples, the ranger travels the same distance back, but the boat travels 5 miles per hour slower than it did on the way up the river. If the ranger spent 3 hours travelling up and back, which of the following equations could be used to determine the speed of the boat on the way up the river? Choose 1 answer: (A) 2v^(2)-13 v+17=0 (B) 3v^(2)-15 v-4=0 (c) 3v^(2)-11 v+10=0 (D) 3v^(2)-19 v+10=0

Question

A forest ranger travels up a river by boat to collect soil samples. The boat travels at a speed of  v miles per hour for a distance of 2 miles. After collecting the samples, the ranger travels the same distance back, but the boat travels 5 miles per hour slower than it did on the way up the river. If the ranger spent 3 hours travelling up and back, which of the following equations could be used to determine the speed of the boat on the way up the river? Choose 1 answer: (A)  2v^(2)-13 v+17=0 (B)  3v^(2)-15 v-4=0 (c)  3v^(2)-11 v+10=0 (D)  3v^(2)-19 v+10=0

Bytelearn · Accepted Answer

Denote Speed and Distance: Let's denote the speed of the boat on the way up the river as v miles per hour. The distance traveled up the river and back is the same, which is 2 miles each way. The speed on the way back is (v - 5) miles per hour because it's 5 miles per hour slower. We need to find the total time spent for both trips to be 3 hours. The time taken to travel a certain distance is equal to the distance divided by the speed. So, the time taken to go up the river is 2/v hours, and the time taken to come back is 2/(v - 5) hours. The sum of these times should equal 3 hours.Set Up Equation: We can set up the equation as follows: Time up the river + Time back = Total time 2/v + 2/(v - 5) = 3 To solve for v, we need to find a common denominator and combine the terms on the left side of the equation.Find Common Denominator: The least common denominator (LCD) for v and (v - 5) is v(v - 5). We multiply each term by the LCD to clear the fractions: v(v - 5)(2/v) + v(v - 5)(2/(v - 5)) = v(v - 5)(3) This simplifies to: 2(v - 5) + 2v = 3v(v - 5)Simplify Equation: Now we distribute and combine like terms: 2v - 10 + 2v = 3v^2 - 15v 4v - 10 = 3v^2 - 15v To set the equation to zero, we bring all terms to one side: 3v^2 - 15v - 4v + 10 = 0Combine Like Terms: Combine the v terms: 3v^2 - 19v + 10 = 0 This is a quadratic equation in standard form, which can be used to solve for v, the speed of the boat on the way up the river.

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