A hot-air balloon, headed due east at an average speed of 15 miles per hour and at a constant altitude of 100 feet, passes over an intersection. Find an ex[ression for the distance d (measured in feet) from ballon to the intersection t seconds later.
1'000 FEET HIGH
2200feet
To find the expression for the distance of the balloon from the intersection, you can use the equation of motion.
First, let's consider the balloon's horizontal (eastward) motion. Since it is moving due east at a constant speed of 15 miles per hour, the distance it covers horizontally is given by the formula:
distance = speed * time
In this case, the speed is given as 15 miles per hour. To convert this to feet per second, we can multiply it by a conversion factor:
1 mile = 5,280 feet
So, 15 miles per hour = 15 * 5,280 feet per hour
To convert from hours to seconds, we divide by 3600 (since there are 3600 seconds in an hour):
15 * 5,280 feet per hour = (15 * 5,280) / 3600 feet per second
Simplifying this expression, we get:
distance = (15 * 5,280) / 3600 * t
Now, let's consider the balloon's constant altitude of 1,000 feet. The vertical distance from the balloon to the ground remains the same, regardless of time.
Therefore, the final expression for the distance from the balloon to the intersection t seconds later is:
d = sqrt((15 * 5,280 / 3600 * t)^2 + 100^2)
Alternatively, you can simplify this expression by squaring both the terms under the square root sign and combining them, then taking the square root. This would give you:
d = sqrt(((15 * 5,280) / 3600)^2 * t^2 + 100^2)
Simplifying further:
d = sqrt((33^2 * t^2 + 100^2))
Therefore, the expression for the distance d (measured in feet) from the balloon to the intersection t seconds later is sqrt(33^2 * t^2 + 100^2).