Answered

Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

A boat can travel at an average speed of 10 miles per hour in still water. Traveling with the current, it can travel 6 miles in the same amount of time as upstream.

Use the relationship [tex]$t = \frac{d}{r}[tex]$[/tex] to construct a rational equation that can be solved for [tex]$[/tex]x$[/tex] to find the speed of the current.

Sagot :

GinnyAnswer
To solve this problem, we'll derive a rational equation to find the speed of the current. Given that:
- The speed of the boat in still water is \( 10 \) miles per hour.
- The distance traveled downstream and upstream is \( 6 \) miles.
- The times taken for both journeys (downstream and upstream) are equal.

Let's denote the speed of the current as \( x \) miles per hour.

1. Downstream journey: The effective speed of the boat travelling downstream (with the current) is \( 10 + x \) miles per hour.
2. Upstream journey: The effective speed of the boat travelling upstream (against the current) is \( 10 - x \) miles per hour.
3. The time to travel a given distance \( d \) at speed \( r \) is given by: \( t = \frac{d}{r} \).

Given that the times for downstream and upstream journeys are equal, we can set their time equations equal to each other:

[tex]\[ \text{Time downstream} = \text{Time upstream} \][/tex]

Expressing these times using the given distance and the boat's effective speeds:
[tex]\[ \frac{6}{10 + x} = \frac{6}{10 - x} \][/tex]

To construct and solve the rational equation, follow these steps:

1. Set up the equation:
[tex]\[ \frac{6}{10 + x} = \frac{6}{10 - x} \][/tex]

2. Eliminate the fractions by multiplying both sides by the denominators (first multiply both sides by \((10 + x)(10 - x)\)):
[tex]\[ 6(10 - x) = 6(10 + x) \][/tex]

3. Simplify the equation:
[tex]\[ 60 - 6x = 60 + 6x \][/tex]

4. Combine like terms:
[tex]\[ 60 - 60 = 6x + 6x \][/tex]
[tex]\[ 0 = 12x \][/tex]

5. Solve for \( x \):
[tex]\[ x = 0 \][/tex]

Thus, the speed of the current \( x \) is 0 miles per hour.

So the rational equation that can be used to find the speed of the current is:
[tex]\[ \frac{6}{10 + x} = \frac{6}{10 - x} \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.