Two butterflies simultaneously leave an aster flowerbed and fly to a rose flowerbed. One butterfly flies 10 m/min faster than the other, and so lands on a rose 1 minute earlier. Find the speed of each butterfly if the flowerbeds are 560m apart
The Correct Answer and Explanation is :
The Correct Answer is :Slower butterfly: 70 m/min
faster butterflies:80m/min
To solve the problem, we can define the speeds of the two butterflies and use the information provided to set up equations.
Step 1: Define Variables
Let:
- ( v ) = speed of the slower butterfly in meters per minute (m/min)
- ( v + 10 ) = speed of the faster butterfly in m/min (since it flies 10 m/min faster)
Step 2: Set Up the Equations
The distance between the flowerbeds is 560 meters. We can express the time taken by each butterfly to reach the rose flowerbed as follows:
- Time taken by the slower butterfly:
[
t_1 = \frac{560}{v}
] - Time taken by the faster butterfly:
[
t_2 = \frac{560}{v + 10}
]
According to the problem, the faster butterfly arrives 1 minute earlier than the slower butterfly:
[
t_1 – t_2 = 1
]
Step 3: Substitute and Simplify
Substituting the expressions for ( t_1 ) and ( t_2 ) into the equation:
[
\frac{560}{v} – \frac{560}{v + 10} = 1
]
To eliminate the fractions, multiply through by ( v(v + 10) ):
[
560(v + 10) – 560v = v(v + 10)
]
[
5600 = v^2 + 10v
]
Step 4: Rearrange into a Quadratic Equation
Rearranging the equation gives:
[
v^2 + 10v – 5600 = 0
]
Step 5: Solve the Quadratic Equation
We can use the quadratic formula:
[
v = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
where ( a = 1 ), ( b = 10 ), and ( c = -5600 ):
[
v = \frac{-10 \pm \sqrt{10^2 – 4 \cdot 1 \cdot (-5600)}}{2 \cdot 1}
]
[
v = \frac{-10 \pm \sqrt{100 + 22400}}{2}
]
[
v = \frac{-10 \pm \sqrt{22500}}{2}
]
[
v = \frac{-10 \pm 150}{2}
]
Calculating the positive root:
[
v = \frac{140}{2} = 70 \, \text{m/min}
]
Step 6: Calculate Speeds
- Speed of the slower butterfly: ( v = 70 \, \text{m/min} )
- Speed of the faster butterfly: ( v + 10 = 80 \, \text{m/min} )
Conclusion
Thus, the speeds of the butterflies are:
- Slower butterfly: 70 m/min
- Faster butterfly: 80 m/min
In conclusion, using algebraic methods and the relationship between distance, speed, and time, we can determine the speeds of the two butterflies based on the conditions given in the problem.