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A motorboat travels 9 miles downstream (with the current) in 90 minutes.

Which system of equations can be used to find [tex]\( x \)[/tex], the speed of the boat in miles per hour, and [tex]\( y \)[/tex], the speed of the current in miles per hour? Recall the formula [tex]\( d = rt \)[/tex].

A. [tex]\( 9 = 0.5(x - y) \)[/tex]
B. [tex]\( 9 = 1.5(x + y) \)[/tex]
C. [tex]\( 9 = 1.5(x - y) \)[/tex]
D. [tex]\( 9 = 0.5(x + y) \)[/tex]
E. [tex]\( 0.5 = 9(x - y) \)[/tex]
F. [tex]\( 1.5 = 9(x + y) \)[/tex]
G. [tex]\( 1.5 = 9(x - y) \)[/tex]
H. [tex]\( 0.5 = 9(x + y) \)[/tex]

Sagot :

Let's approach this step-by-step to determine which system of equations can be used to find [tex]\( x \)[/tex], the speed of the boat in miles per hour, and [tex]\( y \)[/tex], the speed of the current in miles per hour.

1. Understand the problem:
- The boat travels downstream with the current.
- Distance traveled downstream: 9 miles.
- Time taken for the downstream journey: 90 minutes (which is 1.5 hours).

2. Define variables:
- Let [tex]\( x \)[/tex] be the speed of the boat in still water (miles per hour).
- Let [tex]\( y \)[/tex] be the speed of the current (miles per hour).

3. Determine the effective speed:
- When the boat is traveling downstream, its effective speed is [tex]\( (x + y) \)[/tex] because the current aids the boat.

4. Use the distance formula:
- The formula for distance [tex]\( d = r \cdot t \)[/tex], where [tex]\( r \)[/tex] is the rate (or speed) and [tex]\( t \)[/tex] is the time.
- Given [tex]\( d = 9 \)[/tex] miles and [tex]\( t = 1.5 \)[/tex] hours, we can set up the equation for downstream travel:
[tex]\[ 9 = 1.5 \cdot (x + y) \][/tex]

5. Formulate the equation:
- Rearrange the equation to solve for [tex]\( (x + y) \)[/tex]:
[tex]\[ x + y = \frac{9}{1.5} \][/tex]
[tex]\[ x + y = 6 \][/tex]
- However, since we are comparing to options provided:

- The correct equation directly relating distance, speed, and time is:
[tex]\[ 9 = 1.5 \cdot (x + y) \][/tex]

From the given multiple-choice options, the correct one that matches our derived equation is:
[tex]\[ 9 = 1.5(x + y) \][/tex]

Hence, the correct answer is:
[tex]\[ 9 = 1.5(x + y) \][/tex]
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