Question

two vehicles, a car and a truck, leave an intersection at the same time. the car heads east at an average speed of 45 miles per hour, while the truck heads south at an average speed of 24 miles per hour. find an expression for their distance apart d (in miles) at the end of t hours. d = □ mi (simplify your answer.)

Explanation:

Step1: Calculate distance traveled by car

The car's speed is 45 miles per hour. Using the formula $d = vt$ (distance = speed×time), the distance traveled by the car in $t$ hours is $45t$ miles.

Step2: Calculate distance traveled by truck

The truck's speed is 24 miles per hour. Using the formula $d = vt$, the distance traveled by the truck in $t$ hours is $24t$ miles.

Step3: Apply Pythagorean theorem

Since the car moves east and the truck moves south (perpendicular directions), the distance $d$ between them forms the hypotenuse of a right - triangle. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 45t$ and $b = 24t$. So $d^{2}=(45t)^{2}+(24t)^{2}$.

Step4: Simplify the expression

First, expand $(45t)^{2}=45^{2}t^{2}=2025t^{2}$ and $(24t)^{2}=24^{2}t^{2}=576t^{2}$. Then $d^{2}=2025t^{2}+576t^{2}=2601t^{2}$. Taking the square root of both sides, $d=\sqrt{2601t^{2}}$. Since $t\geq0$, $d = 51t$.

Answer:

$51t$