A hot-air balloon, headed due east at an average speed of v = 14 miles per hour and at a constant altitude of h = 120 feet, passes over an intersection (see the figure). Find an expression for its distance d (measured in feet) from the intersection t seconds later.

The balloon travels 14mi/hr*t/3600 where t is in seconds. I suppose it would be better to change the velocity to ft/sec

Memorize this conversion factor 60mph=88fps

the balloon travels then 14/60*88 t feet.

the triangle then yields a slant height of
distance= sqrt (120^2+(14*88/60 t)^2 )

To find an expression for the distance of the hot-air balloon from the intersection t seconds later, we need to consider the horizontal motion of the balloon.

The formula for distance is given by:
Distance = Speed × Time

In this case, the speed of the balloon (v) is 14 miles per hour. However, the distance needs to be expressed in feet, so we need to convert miles to feet. 1 mile is equal to 5,280 feet, so the speed in feet per hour is:

Speed = 14 miles/hour × 5,280 feet/mile = 73,920 feet/hour

Now, we want to find the distance in feet, not per hour. We can convert hours to seconds by dividing by 3,600 (since there are 3,600 seconds in an hour):

Speed = 73,920 feet/hour ÷ 3,600 seconds/hour = 20.533 feet/second (approximately)

Therefore, the expression for the distance of the hot-air balloon from the intersection t seconds later is:

d(t) = 20.533 feet/second × t seconds

So, the distance d(t) at any given time t (in seconds) is equal to 20.533 times t.