SOLUTION: The wind is blowing at a steady rate of 10 mph. It takes a truck a total of 5 hours to travel 120 miles west against the wind and then 120 miles back with the wind. What is the t

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Question 224164: The wind is blowing at a steady rate of 10 mph. It takes a truck a total of 5 hours to travel 120 miles west against the wind and then 120 miles back with the wind. What is the trucks speed in still air?
Found 2 solutions by Alan3354, nerdybill:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The wind is blowing at a steady rate of 10 mph. It takes a truck a total of 5 hours to travel 120 miles west against the wind and then 120 miles back with the wind. What is the trucks speed in still air?
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The truck travels 240 miles in 5 hours
240/5 = 48 mph
I don't think the wind has any effect.

Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
The wind is blowing at a steady rate of 10 mph. It takes a truck a total of 5 hours to travel 120 miles west against the wind and then 120 miles back with the wind. What is the trucks speed in stil
.
Applying the distance formula: d = rt
.
Let x = speed of truck in still wind
and y = time traveling against the wind
then
"distance traveling against wind"
(x-10)y = 120
"distance traveling with wind"
(x+10)(10-y) =120
.
So, now we have two equations and two unknowns:
(x-10)y = 120
(x+10)(10-y) =120
.
Our system of equations:
(x-10)y = 120 (equation 1)
.
(x+10)(10-y) =120
10x-xy-10y+100 = 120 (equation 2)
.
Solve equation 1 for y:
y = 120/(x-10)
.
Plug the definition of y (above) into equation 2 and solve for x:
10x-xy-10y+100 = 120
10x+100-xy-10y = 120
10x+100-(xy+10y) = 120
10x+100-y(x+10) = 120
10x+100-(120/(x-10))(x+10) = 120
Multiply BOTH sides by (x-10) to get rid of denominators:
10x(x-10)+100(x-10)-120(x+10) = 120(x-10)
10x^2-100x+100x-1000-120x-1200 = 120x-1200
10x^2-120x-2200 = 120x-1200
10x^2-240x-1000 = 0
x^2-24x-100 = 0
Since we can't factor, we must resort to the quadratic equation. Doing so will yield:
x = {27.62, -3.62}
A negative speed does NOT make sense, so toss it out leaving:
x = 27.62 mph
.
Details of quadratic equation to follow:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-24x%2B-100+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-24%29%5E2-4%2A1%2A-100=976.

Discriminant d=976 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--24%2B-sqrt%28+976+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-24%29%2Bsqrt%28+976+%29%29%2F2%5C1+=+27.6204993518133
x%5B2%5D+=+%28-%28-24%29-sqrt%28+976+%29%29%2F2%5C1+=+-3.62049935181331

Quadratic expression 1x%5E2%2B-24x%2B-100 can be factored:
1x%5E2%2B-24x%2B-100+=+1%28x-27.6204993518133%29%2A%28x--3.62049935181331%29
Again, the answer is: 27.6204993518133, -3.62049935181331. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-24%2Ax%2B-100+%29