Chapter 1.1, Problem 63AYU

Distance between Moving Objects A Ford Focus and a Freightliner Cascadia truck leave an intersection at the same time. The Focus heads east at an average speed of $60$ miles per hour, while the Cascadia heads south at an average speed of $45$ miles per hour. Find an expression for their distance apart $d$(in miles) at the end of $t$ hours.

To determine

To calculate: An expression for the distance $d$ (in miles) at the end of $t$ hours between the Ford Focus and Freightliner truck.

Answer to Problem 63AYU

Solution:

An expression for the distance $d$ (in miles) at the end of $t$ hours between the Ford Focus and Freightliner truck is $d=75t\text{miles}$

Explanation of Solution

Given Information:

The focus heads east at an average speed of $60$ miles per hour.

The Freightliner Cascadia truck heads south at an average speed of $45$ miles per hour

Formula used:

1) The relation between distance, speed and time. $\text{Distance}\text{=}\text{speed}\text{×}\text{time}$

2) The distance formula: The distance between two points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ denoted by $d\left(P_1,P_2\right)$, is $d\left(P_1,P_2\right)=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$.

Calculation:

To find the distance of Ford focus and Freightliner truck first find distance covered by Ford focus and Freightliner by using the relation between distance, speed and time $\text{Distance}\text{=}\text{speed}\text{×}\text{time}$ in $t$ hours.

The Ford focus covered $60t$ miles, so $P_1=(60t,0)$ and Freightliner truck covered $-45t$ so $P_2=(0,-45t)$(negative sign indicates south direction)

Consider the graph below

To find the distance $d$ (in miles) at the end of $t$ hours between the Ford Focus and Freightliner truck that means from the point $P_1=(60t,0)$ and $P_2=(0,-45t)$

By using the distance formula the distance between two points $P_1=(60t,0)$ and $P_2=(0,-45t)$

$d=\sqrt{{(0-60t)}^2+{(-45t-0)}^2}$

$d=\sqrt{{(-60t)}^2+{(-45t)}^2}$

$=\sqrt{3600t^2+2025t^2}$

$=\sqrt{5625t^2}$

$=75t$

Therefore, an expression for the distance $d$ (in miles) at the end of $t$ hours between the Ford Focus and Freightliner truck is $d=75t\text{miles}$