Chapter 1.1, Problem 64AYU

Distance of a Moving Object from a Fixed Point A hot-air balloon, headed due east at an average speed of 15 miles per hour at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for its distance $d$ (measured in feet) from the intersection $t$ seconds later.

To determine

To find: An expression for the distance of a hot-air balloon (in feet) from the intersection it passes, t seconds later.

Answer to Problem 64AYU

Distance $=\text{d}=\sqrt{225t^2 + 10000}\text{feet}$.

Explanation of Solution

Given:

Average speed of hot air balloon $=15\text{ miles/hour}$.

Altitude of hot air balloon $=100\text{ feet}$.

Formula used:

$\text{Distance = speed × time}$

Distance formula $=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$.

Calculation:

Let distance travelled by balloon $t$ seconds later at a height of $100\text{ feet}$ horizontally $=15t\left(\text{speed × time}\right)$.

Then distance from the point of intersection on the ground $=d=\sqrt{{(15t-0)}^2+{(100-0)}^2}$.

$=\sqrt{{(15t)}^2+{(100)}^2}$

$=\sqrt{225t^2+10000}$

Thus distance from intersection t seconds later $=\sqrt{225t^2+10000}$.