Question

A truck's varying distance from the warehouse, d , measured in miles is modeled by the formula d=t2+50t where t is the varying number of hours since the truck left the warehouse. The truck's destination is 437.5 miles away.

a.How many hours will it take the truck to reach its destination? (Hint: You may need to use your graphing calculator to answer this question).

b.A car left the warehouse at the same time as the truck and wants to get to the destination at the same time. What constant speed would the car have to travel in order to reach the destination at the same time as the truck?

Answer 1

• 7.596 hours
• 57.596 mph

Explanation:

a) The distance is given as ...

d = t^2 +50t

We want to find the value of t for which this distance is 437.5 miles. That means we want to solve the equation ...

437.7 = t^2 +50t

We can complete the square by adding the square of half the t coefficient, (50/2)^2 = 625

437.5 +625 = t^2 +50t +625

1062.5 = (t +25)^2

√1062.5 -25 = t ≈ 7.596 . . . . hours

It will take the truck about 7.596 hours to reach its destination.

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b) Average speed is total distance divided by total time. Here, we can find the average speed as ...

d/t = (t^2 +50t)/t = t +50

That is, the average speed over the 437.5-mile trip is ...

7.596 +50 = 57.596 . . . . miles per hour

The car must maintain a constant speed of 57.596 mph to reach the destination at the same time.

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When solving equations using a graphing calculator, I like to put them into a form such that the solution is the x-intercept. Here, that means the equation we're solving is ...

x^2 +50x -437.5 = 0 . . . . x is hours to destination