Problem 40 - Elementary Algebra

Question

For the following problems, the given trinomial occurs when solving the corresponding applied problem. Factor each trinomial. You do not need to solve the problem. \(5 r^{2}-24 r-5\) It takes 5 hours to paddle a boat 12 miles downstream and then back. The current flows at the rate of 1 mile per hour. At what rate was the boat paddled?

Step-by-Step Solution

Answer

Answer: The boat is paddled at a rate of 5 miles per hour in still water.

Step 1: Factor the trinomial

We have the trinomial \(5r^2 - 24r -5\). To factor this, we can look for two binomials that when multiplied with each other will give this trinomial. In our case, the factored form is \((5r + 1)(r - 5)\).

Step 2: Analyze the downstream and upstream journey

Let \(r\) be the paddling rate of the boat in still water, and let \(c=1\) mph be the rate of the current. When the boat is paddled downstream, the rate is \((r+c)\) mph, and when paddled upstream, the rate is \((r-c)\) mph. The total distance covered downstream and upstream is 12 miles, and the total time taken is 5 hours. So, we can create two equations: 1. Downstream: Distance = Rate × Time \((12 = (r+1)t_1)\) 2. Upstream: Distance = Rate × Time \((12 = (r-1)t_2)\) Since the total time is 5 hours, we have \(t_1 + t_2 = 5\).

Step 3: Relate the equations with the factored trinomial

We know the factored form of the trinomial is \((5r + 1)(r - 5)\). The second binomial in the factored form corresponds to the time taken to cover the distance in upstream, which is \(t_2 = (r - 5)\). Now, we can find the expression for \(t_1\) by substituting the value of \(t_2\) in the equation \(t_1 + t_2 = 5\): \(t_1 = 5 - (r - 5)\) \(t_1 = 10 - r\)

Step 4: Find the rate at which the boat is paddled

To find the paddling rate of the boat, we need to find the value of \(r\) that satisfies the equation \((5r + 1)(r - 5) = 0\). Applying the Zero-Product Property, we have two cases: 1. \((5r + 1) = 0\) ⟹ \(r = -\frac{1}{5}\) (This is not possible as the paddling rate cannot be negative) 2. \((r - 5) = 0\) ⟹ \(r = 5\) So, the boat is paddled at a rate of 5 miles per hour.

Key Concepts

Paddling Rate

The paddling rate refers to how fast a boat can move through still water, excluding the effect of any current. In this particular exercise, the variable \(r\) is used to represent the paddling rate. This is a crucial factor when calculating travel time while paddling either downstream or upstream.

• Downstream Rate: When going downstream, the current aids the boat, increasing its speed. The effective rate is \((r+c)\), where \(c\) is the current's speed.
• Upstream Rate: Conversely, moving upstream means fighting the current, reducing the speed to \((r-c)\).

Understanding paddling rate is key to solving problems that involve river currents, where one must calculate how long a trip will take based on these variables. It’s vital for determining travel schedules when the exact timing of downstream and upstream journeys is essential.

Zero-Product Property

The Zero-Product Property is a fundamental algebraic concept used to solve equations where the product of two factors equals zero. This property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). This comes into play when factoring trinomials to solve for variables.In our exercise, once the trinomial \(5r^2 - 24r - 5\) was factored into \((5r + 1)(r - 5)\), we applied the Zero-Product Property:

• Set \(5r + 1 = 0\) which solves to \(r = -\frac{1}{5}\), and
• Set \(r - 5 = 0\) which solves to \(r = 5\).

Since a negative paddling rate doesn't make sense in this context, we discard \(-\frac{1}{5}\) and accept \(r = 5\) miles per hour as the practical solution. This shows how crucial the Zero-Product Property is, as it narrows down potential solutions by excluding those that aren't feasible.

Downstream and Upstream Problems

Downstream and upstream problems are classic types of word problems involving travel against and along current, breeze, or other forces. The downstream direction moves with the current, thus the effective speed increases, whereas the upstream direction moves against the current, reducing effective speed.In this problem, key components include:

• Distance: The exercise involved a total round-trip distance of 24 miles (12 miles each way).
• Time: The complete journey takes 5 hours, split based on downstream (\(t_1\)) and upstream (\(t_2\)) travel times.
• Equations: \(12 = (r+1)t_1\) for downstream and \(12 = (r-1)t_2\) for upstream.

The solution relied on setting up these two equations and using the total time constraint \(t_1 + t_2 = 5\). By investigating these relationships, one can understand the broader concept of how forces acting on a vessel affect travel times and speeds, and also learn to set up equations that balance these variables together.