Chapter 10.7, Problem 57AYU

Uniform Motion AToyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection. See the figure.

a. Find parametric equations that model the motion of the Camry and Impala.

b. Find a formula for the distance between the cars as a function of time.

c. Graph the function in part (b) using a graphing utility.

d. What is the minimum distance between the cars? When are the cars closest?

e. Simulate the motion of the cars by simultaneously graphing the equations found in part (a).

To determine

To find:

a. Parametric equations that model the motion of the Camry and Impala.

Answer to Problem 57AYU

a. Camry: $x = 40t - 5$, $y = 0$; Impala: $x = 0$, $y = 30t - 4$

Explanation of Solution

Given:

A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection.

Calculation:

a. At $t = 0$,

Camry is at a distance of 5 miles from intersection and is travelling east at a speed of 40mph.

Parametric equation for Camry:

$x = 40t - 5$, $y = 0$

At $t = 0$,

Impala is 4 miles from the intersection and is travelling north at a speed of 30mph.

Parametric equation for Impala:

$x = 0$, $y = 30t - 4$

To determine

To find:

b. Formula for the distance between the cars as a function of time?

Answer to Problem 57AYU

b. $d = \sqrt{\left(40t - 5\right)² + {\left(30t - 4\right)}^2}$

Explanation of Solution

Given:

A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection.

Calculation:

b. As per the given figure we can use Pythagoras theorem to find the distance $d$ in terms of $t$

$d = \sqrt{\left(40t - 5\right)² + {\left(30t - 4\right)}^2}$

To determine

To find:

c. Graph the function in part c.

Answer to Problem 57AYU

c. graph is plotted

Explanation of Solution

Given:

A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection.

Calculation:

c. Graph the function in part c.

To determine

To find:

d. The minimum distance between the cars? When are they closest?

Answer to Problem 57AYU

d. distance between the cars is $0.2$ miles. At $0.128$ hours, the cars are closer to each other.

Explanation of Solution

Given:

A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection.

Calculation:

d. From the graph above the minimum distance between the cars is $0.2$ miles. At $0.128$ hours, the cars are closer to each other.

To determine

To find:

e. Simulate the motion of the cars by simultaneously graphing the equations found in part (a).

Answer to Problem 57AYU

e. Graph is plotted.

Explanation of Solution

Given:

A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection.

Calculation:

e. Graph of the equation found in a.