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 time it takes to travel up the river is the distance divided by the speed, which is 2/v hours. On the way back, the boat is traveling 5 miles per hour slower, so the speed is v-5 miles per hour. The time it takes to travel back is the distance divided by this slower speed, which is 2/(v-5) hours. The total time spent traveling is the sum of these two times, which is given as 3 hours. Therefore, we can set up the equation: 2/v + 2/(v-5) = 3Set Up Equation: Now we need to solve this equation for v. To do this, we need to find a common denominator for the two fractions on the left side of the equation. The common denominator is v(v-5). We multiply both sides of the equation by this common denominator to eliminate the fractions: v(v-5)(2/v) + v(v-5)(2/(v-5)) = 3v(v-5)Solve for v: Simplifying the equation, we get: 2(v-5) + 2v = 3v^2 - 15v This simplifies to: 2v - 10 + 2v = 3v^2 - 15vCommon Denominator: Combining like terms on the left side gives us: 4v - 10 = 3v^2 - 15v Now we want to set the equation to zero by moving all terms to one side: 3v^2 - 15v - 4v + 10 = 0Simplify Equation: Simplifying the equation further, we combine the v terms: 3v^2 - 19v + 10 = 0 This is a quadratic equation in standard form.Combine Like Terms: Looking at the answer choices, we see that the equation we derived, 3v^2 - 19v + 10 = 0, matches choice (D).

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