Suppose two cars depart from a four-way intersection at the same time, one heading north and the other heading west. The car heading north travels at the steady speed of 20 ft/sec and the car heading west travels at the steady speed of 36 ft/sec.
(a) Find an expression for the distance between the two cars after t seconds. (Round your coefficients to one decimal place as needed.) in ft
(b) Find the distance in miles between the two cars after 3 hours 15 minutes. (Round your answer to one decimal place.) in mi
(c) When are the two cars 1 mile apart? (Round your answer to one decimal place.)in sec
since distance = speed * time,
(a) d = √((20t)^2 + (36t)^2) = 4√106 t
(b) use t=3*3600+15*60 = 11,700 in (a) and divide by 5280
(c) solve for t when d = 5280
(a) The car heading north is traveling at a speed of 20 ft/sec, so its position can be represented by the equation x = 20t, where x is the distance traveled in feet and t is the time in seconds.
Similarly, the car heading west is traveling at a speed of 36 ft/sec, so its position can be represented by the equation y = 36t, where y is the distance traveled in feet and t is the time in seconds.
To find the distance between the two cars, we can use the Pythagorean theorem, which states that the square of the hypotenuse (distance between the two cars) is equal to the sum of the squares of the other two sides (distances traveled by each car).
Distance^2 = x^2 + y^2
Substituting the given equations for x and y, we get:
Distance^2 = (20t)^2 + (36t)^2
Distance^2 = 400t^2 + 1296t^2
Distance^2 = 1696t^2
Taking the square root of both sides, we find:
Distance = sqrt(1696t^2)
Distance = 41.2t
Therefore, the expression for the distance between the two cars after t seconds is 41.2t feet.
(b) To find the distance in miles after 3 hours and 15 minutes, we first need to convert the time into seconds.
3 hours = 3 * 60 * 60 = 10800 seconds
15 minutes = 15 * 60 = 900 seconds
Total time = 10800 + 900 = 11700 seconds
Now, we can substitute the given time into the expression we found in part (a):
Distance = 41.2 * 11700
Distance ≈ 481,140 feet
To convert this distance into miles, we divide by the number of feet in a mile:
Distance in miles = 481,140 / 5280 ≈ 91.2 miles
Therefore, the distance between the two cars after 3 hours 15 minutes is approximately 91.2 miles.
(c) To find when the two cars are 1 mile apart, we set the distance equal to 1 mile and solve for t:
1 = 41.2t
t = 1 / 41.2
t ≈ 0.0244 seconds
Therefore, the two cars are approximately 1 mile apart after 0.0244 seconds.
To find the distance between the two cars after a certain time, we can use the formula:
Distance = Speed * Time
(a) Let's assume the car heading north is at position (0, 0) initially, and the car heading west is at position (-t, 0) initially. The distance between the two cars can be calculated as the hypotenuse of a right triangle formed by the two cars.
Using the Pythagorean theorem, the distance between the two cars after t seconds can be calculated as:
Distance = sqrt((20t)^2 + (36t)^2)
Simplifying this expression, we get:
Distance = sqrt(400t^2 + 1296t^2)
Distance = sqrt(1696t^2)
Distance = 41.2t (rounded to one decimal place)
Therefore, the expression for the distance between the two cars after t seconds is 41.2t ft.
(b) To find the distance between the two cars after 3 hours 15 minutes, we need to convert the time to seconds first.
3 hours 15 minutes = 3 * 60 * 60 + 15 * 60 = 11,700 seconds
Now we can substitute this value into the expression we derived in part (a):
Distance = 41.2 * 11,700
Distance = 481,440 ft
To convert this distance to miles, divide it by the number of feet in a mile:
Distance in miles = 481,440 ft / 5,280 ft/mi
Distance in miles = 91.23 mi (rounded to one decimal place)
Therefore, the distance between the two cars after 3 hours 15 minutes is 91.2 mi.
(c) To find when the two cars are 1 mile apart, we can set the distance equal to 1 mile and solve for t:
1 = 41.2t
Simplifying for t, we get:
t = 1 / 41.2
t ≈ 0.0243 seconds (rounded to one decimal place)
Therefore, the two cars are approximately 1 mile apart after 0.0243 seconds.
To find the distance between the two cars after t seconds, we need to consider their relative positions. Let's assume that the car heading north starts from the origin (0, 0), and the car heading west starts from the point (0, 0) as well.
(a) The car heading north will move only in the y-direction at a constant speed of 20 ft/sec. Hence, its position can be represented by the coordinates (0, 20t).
The car heading west will move only in the x-direction at a constant speed of 36 ft/sec. Thus, its position can be represented by the coordinates (36t, 0).
To find the distance between the two cars after t seconds, we can use the distance formula, which is sqrt((x2 - x1)^2 + (y2 - y1)^2).
So, the distance d between the two cars after t seconds is:
d = sqrt((36t - 0)^2 + (20t - 0)^2)
d = sqrt(1296t^2 + 400t^2)
d = sqrt(1696t^2)
d = 41.2t
Therefore, the expression for the distance between the two cars after t seconds is 41.2t ft.
(b) To find the distance in miles after 3 hours 15 minutes (which is 3.25 hours), we need to convert the time from hours to seconds.
3 hours = 3 * 60 * 60 = 10800 seconds
15 minutes = 15 * 60 = 900 seconds
Total time t = 10800 + 900 = 11700 seconds
Substituting this value into the expression we obtained in part (a):
d = 41.2t
d = 41.2 * 11700
d ≈ 481,080 ft
To convert this distance from feet to miles, we need to divide by 5280 (since there are 5280 feet in one mile):
d ≈ 481,080 / 5280 ≈ 91.2 miles
Therefore, the distance between the two cars after 3 hours 15 minutes is approximately 91.2 miles.
(c) To find when the two cars are 1 mile apart, we need to set the distance (d) equal to 1 mile and solve for t.
1 mile = 5280 feet
Setting 41.2t = 5280:
41.2t = 5280
t = 5280 / 41.2
t ≈ 128.16 seconds
Therefore, the two cars are approximately 1 mile apart after 128.2 seconds.