1) Matt traveled the first leg of his trip for 1350 miles on a large jet. He then traveled an additional 600 miles on a private plane to reach his destination. If the speed of the jet was three times the speed of the plane and the total time in the air was 6 hours determine the speed of each aircraft.
2) Matt and Sara mow lawns together. When Matt mows the lawn for Mr. Smith it takes 100 minutes. When matt and Sara work together to mow the lawn for Smith they get the job done in 40 minutes. How long will it take Sara to mow Smith's lawn by herself?
speed of private plane --- x mph
speed of jet -----> 3x mph
time for 1st leg of trip = 1350/x
time for 2nd leg of trip = 600/(3x) = 200/x
so 1350/x + 200/x = 6
1550/x=6
6x = 1550
x = 258.333..
private plane went 258.33 mph, jet went 775 mph
Matt's rate = lawn/100
Sara's rate = lawn/x , where x is Sara's time if she works alone
combined rate = lawn/100 + lawn/x
= (xlawn + 100lawn)/(100x)
= lawn*(x+100)/(100x)
then lawn*(x+100)/(100x) = lawn/40
(x+100)/(100x) = 1/40
cross-multiply
100x = 40x + 4000
60x = 4000
x = 66.666. or 66 2/3 minutes
To solve these problems, we can use the formulas:
Distance = Speed * Time
Speed = Distance / Time
Time = Distance / Speed
Let's use these formulas to find the speeds of the aircraft in the first problem and the time it will take Sara to mow the lawn in the second problem.
1) Let's assume the speed of the private plane is x miles per hour. According to the problem, the speed of the large jet is three times the speed of the private plane, which means it's 3x miles per hour.
The total time in the air was 6 hours, so we can set up two equations:
1350 miles = 3x * t1 (where t1 is the time it took for the large jet to travel the first leg)
600 miles = x * t2 (where t2 is the time it took for the private plane to travel the additional distance to the destination)
We can rearrange these equations to solve for t1 and t2:
t1 = 1350 / (3x)
t2 = 600 / x
Since the total time in the air was 6 hours, we have the equation:
t1 + t2 = 6
Substituting the values of t1 and t2, we get:
1350 / (3x) + 600 / x = 6
To solve this equation, we need to find the common denominator and combine like terms:
(1350 + 600) / (3x) = 6
1950 / (3x) = 6
Now, we can solve for x:
1950 = 18x
x = 1950 / 18
x = 108.33
The speed of the private plane is approximately 108.33 miles per hour, and the speed of the large jet is three times that, which is approximately 324.99 miles per hour.
2) Let's assume Sara's speed is y lawns per minute. According to the problem, Matt takes 100 minutes to mow the lawn by himself, and together they take 40 minutes to mow the lawn.
We can set up two equations:
1 yard = Matt's speed * 100 minutes
1 yard = (Matt's speed + Sara's speed) * 40 minutes
Simplifying the second equation, we get:
1 yard = (Matt's speed + y) * 40 minutes
Now, we can solve for y (Sara's speed):
40(Matt's speed + y) = 1
40Matt's speed + 40y = 1
40y = 1 - 40Matt's speed
y = (1 - 40Matt's speed) / 40
From the first equation, we know Matt's speed, which is 1 yard per 100 minutes. Substituting this value into the second equation, we get:
y = (1 - 40/100) / 40
y = (1 - 0.4) / 40
y = 0.6 / 40
y = 0.015
Therefore, it will take Sara approximately 0.015 minutes (or 0.9 seconds) to mow Mr. Smith's lawn by herself.