the stopping distance of an automobile is directly proportional to the square of its speed v. a car required 90 feet to stop when its speed was 70 miles per hour. find a mathematical model that gives the stopping distance d in terms of its speed v.
D=
Estimate the stopping distance if the brakes are applied when the car is traveling at 71 miles per hour.
D = k v^2 ... 90 = k * 70^2 ... k = 9 / 490
D = (9/490) * 71^2
Oh, I see what we're dealing with here! We have a classic case of "stopping distance" - the automotive dilemma that causes many drivers to wish they had a teleportation device.
So, let's get our mathematical hands dirty, shall we? According to the given information, the stopping distance (let's call it "D") is directly proportional to the square of the speed ("v") of the car.
Mathematically, this can be represented as:
D = kv^2
Now, to determine the value of our constant "k", we can use the given data point. When the car was cruising at a speed of 70 miles per hour, the stopping distance was 90 feet. Plugging these values into our equation, we get:
90 = k * (70)^2
Solving for "k", we find:
k = 90 / (70^2)
So, our mathematical model for the stopping distance "D" in terms of the speed "v" is:
D = (90 / (70^2)) * v^2
Now, let's put this model to the test! If the car is traveling at 71 miles per hour, we can estimate the stopping distance by plugging this value into our equation:
D = (90 / (70^2)) * (71)^2
Calculating this out will give us our estimated stopping distance. But hey, remember that this is just an estimate, not a crystal ball. So drive safe and stay alert out there!
To find the mathematical model that gives the stopping distance d in terms of speed v, we can use the information given: the stopping distance is directly proportional to the square of the speed.
Let's assign variables to the stopping distance and the speed:
- Let d represent the stopping distance in feet.
- Let v represent the speed in miles per hour.
According to the given information, we know that when the speed is 70 miles per hour, the stopping distance is 90 feet. We can use this to set up a proportion:
(70^2) / 90 = (v^2) / d
Simplifying the equation:
4900 / 90 = (v^2) / d
54.44 = (v^2) / d
Now, we have the mathematical model for the stopping distance in terms of speed:
d = (v^2) / 54.44
To estimate the stopping distance when the car is traveling at 71 miles per hour, we can substitute v = 71 into the equation:
d = (71^2) / 54.44
Simplifying the equation:
d = 5041 / 54.44
d ≈ 92.63 feet
Therefore, the estimated stopping distance when the car is traveling at 71 miles per hour is approximately 92.63 feet.
To find a mathematical model that gives the stopping distance in terms of speed, we can use the given information that the stopping distance is directly proportional to the square of the speed. Therefore, we can write the mathematical model as:
D = kv^2
Where D represents the stopping distance, v represents the speed, and k is a constant of proportionality.
To find the value of k, we can use the given information that the car required 90 feet to stop when its speed was 70 miles per hour. By plugging in these values into the equation, we have:
90 = k * (70)^2
Simplifying the equation:
90 = k * 4900
k = 90 / 4900
k ≈ 0.018
Therefore, our mathematical model becomes:
D = 0.018v^2
Now, to estimate the stopping distance if the brakes are applied when the car is traveling at 71 miles per hour, we can substitute this speed into our model:
D = 0.018(71)^2
D ≈ 90.738 feet
Therefore, the estimated stopping distance would be approximately 90.738 feet.