Chapter 10.7, Problem 57AYU

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.