Navigation. The current in a typical Mississippi River shipping route flows at a rate of 4 mph. In order for a barge to travel 24 mi upriver and then return in a total of 5 hr, approximately how fast must the barge be able to travel in still water?Solve by factoring: 9t2 + 3t= 0

To determine the speed at which the barge must be able to travel in still water, we'll first calculate the speed of the current and then use it to find the speed of the barge.

Let's begin by finding the speed of the current:

Given: the current flows at a rate of 4 mph.

Now, let's calculate the time taken for the barge to travel upstream and downstream:

Given: the barge travels a distance of 24 miles upriver and then returns.

Let the speed of the barge in still water be represented by v.

When the barge is traveling upriver, it is working against the current, so its effective speed will be (v - 4) mph.

Using the formula time = distance / speed, the time taken for the barge to travel upstream is 24 / (v - 4) hours.

When the barge is traveling downstream, it will be aided by the current, so its effective speed will be (v + 4) mph.

The time taken for the barge to travel downstream is also 24 / (v + 4) hours.

Given: the total time taken for the round trip is 5 hours.

According to the given information, the time taken for the upstream journey plus the time taken for the downstream journey is equal to 5 hours:

24 / (v - 4) + 24 / (v + 4) = 5

Now, we can solve this equation to find the speed of the barge in still water:

Let's multiply both sides of the equation by (v - 4)(v + 4) to eliminate the denominators:

[(v - 4)(v + 4)] * [24 / (v - 4) + 24 / (v + 4)] = 5 * (v - 4)(v + 4)

Canceling out the denominators, we have:

24(v + 4) + 24(v - 4) = 5(v - 4)(v + 4)

Simplifying the equation further:

24v + 96 + 24v - 96 = 5(v^2 - 16)

48v = 5v^2 - 80

Rearranging and simplifying the equation:

5v^2 - 48v - 80 = 0

Now, let's solve the quadratic equation by factoring:

Start by looking for two numbers whose product is -400 (the product of the coefficient of v^2 and the constant term -80), and whose sum is -48 (the coefficient of v).

The numbers that satisfy these conditions are -40 and 10.

Therefore, let's rewrite the quadratic equation as:

5v^2 - 40v + 10v - 80 = 0

Taking out the common factors, we have:

5v(v - 8) + 10(v - 8) = 0

Now, we can factor out the common term (v - 8):

(5v + 10)(v - 8) = 0

Setting each factor equal to zero, we get:

5v + 10 = 0 --> 5v = -10 --> v = -2

v - 8 = 0 --> v = 8

Now we have two possible values for v: -2 and 8. However, since we are dealing with speeds, a negative value doesn't make sense in this context, so we can discard v = -2.

Therefore, the speed at which the barge must be able to travel in still water is approximately 8 mph.