two cars are approaching an intersection one is 3 miles south of the intersection and is moving at a constant speed of 20 miles per hour. at the same time the other car is 4 miles east of the intersection and is moving at the constant speed of 30 miles per hour.
a)express the distance d as a function of time t.
b)Use a graphing utility to graph d=d(t). For what value of t is d smallest?
At a time of t hours after the action started,
- distance remaining for the northbound car before reaching intersection is 3 - 20t
- distance remaining for the westbound car before reaching the intersection is 4 - 30t
let d be the distance between them
d^2= (3-20t)^2 + (4-30t)^2
= 9 - 120t + 400t^2 + 16 - 240t + 900t^2
= 1300t^2 - 360t + 25
I have no idea what "graphing utility " you are using.
for a min of d , dd/dt = 0
2d dd/dt = 2600t - 360
2600t = 360
t = .13846... hrs
a) To express the distance d as a function of time t, we can use the Pythagorean theorem. Let's consider the right triangle formed by the two cars and the intersection. The distance d can be calculated using the formula:
d = √((3 + 20t)^2 + (4 + 30t)^2)
Here, t represents the time elapsed since the cars started moving.
b) Using a graphing utility, we can plot the function d = √((3 + 20t)^2 + (4 + 30t)^2). The graph will show the relationship between distance and time. To find the value of t when d is smallest, we need to locate the minimum point on the graph.
a) To express the distance d as a function of time t, we can use the Pythagorean theorem. Let's consider the car that is 3 miles south of the intersection as the vertical side of the triangle, and the car that is 4 miles east of the intersection as the horizontal side of the triangle.
The vertical side of the triangle is given by d1 = 3 + 20t, where t is the time in hours.
The horizontal side of the triangle is given by d2 = 4 + 30t, where t is the time in hours.
Using the Pythagorean theorem, the distance d between the two cars can be found as:
d = sqrt((d1)^2 + (d2)^2)
= sqrt((3 + 20t)^2 + (4 + 30t)^2)
b) To graph d = d(t) using a graphing utility, we can plot the function d as a function of t. The x-axis represents time t, and the y-axis represents the distance d.
To find the value of t at which d is smallest, we can observe the graph and look for the lowest point on the graph. This would be the minimum value of d, and the corresponding value of t would be the value for which d is smallest.
To find the distance of the cars from the intersection as a function of time, we can use the formula for distance: distance = speed * time.
Let's consider the first car, which is 3 miles south of the intersection and moving at a constant speed of 20 miles per hour. Since it is moving directly towards the intersection, its distance from the intersection decreases over time.
For the first car:
Distance from the intersection = 3 miles - (20 miles per hour) * (time in hours)
d(t) = 3 - 20t
Now let's consider the second car, which is 4 miles east of the intersection and moving at a constant speed of 30 miles per hour. Since it is moving perpendicular to the intersection, its distance from the intersection remains constant over time.
For the second car:
Distance from the intersection = 4 miles
d(t) = 4
To graph the function d = d(t), we can use a graphing utility such as Desmos or Graphing Calculator.
Graphing d(t) = 3 - 20t and d(t) = 4 on the same graph will show how the distances change over time.
To find the value of t for which d is smallest, we need to find the minimum point on the graph. This can be done by observing the graph or using calculus, specifically by taking the derivative of the function d(t) and finding the value of t where the derivative equals zero.
Once the graph is plotted, we can visually analyze it or use differentiation techniques to find the value of t at which d is smallest.