Hello! I would be really grateful if you could help me with this word problem.

Here is the problem: In a cartoon, a malfunctioning cannon fires a hungry coyote towards the bottom of a cliff with an initial rate of 100 feet per second. If the cliff is 1250 feet tall, how long will it take the coyote to reach the desert floor? (To account for gravity, use the formula d= rt + 16t^2, where d = distance, r= initial rate, and t = time. )

Thank You so much!

Just substitute into the equation:

1250 = 100t +16t^2

rewrite as 16t^t + 100 t -1250 = 0

Factor and solve for t

Of course! I'd be happy to help you with this word problem.

To find the time it takes for the coyote to reach the desert floor, we can use the given formula: d = rt + 16t^2, where d is the distance, r is the initial rate, and t is the time.

In this problem, we are given the initial rate (r) of the coyote, which is 100 feet per second, and the height of the cliff (d), which is 1250 feet.

We need to solve the equation d = rt + 16t^2 for t, which represents the time needed for the coyote to reach the desert floor.

Substituting the given values into the equation, we get 1250 = 100t + 16t^2.

Now, we can rearrange the equation to form a quadratic equation in standard form: 16t^2 + 100t - 1250 = 0.

To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring may be a little challenging, so we will use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients in the equation.

In our equation, a = 16, b = 100, and c = -1250. Substituting these values into the quadratic formula, we get:

t = (-100 ± √(100^2 - 4 * 16 * -1250)) / (2 * 16).

Now, we can simplify the equation:

t = (-100 ± √(10000 + 80000)) / 32
= (-100 ± √90000) / 32
= (-100 ± 300) / 32.

By solving the two possibilities separately, we get two potential values for t:

t1 = (-100 + 300) / 32
= 200 / 32
= 6.25.

t2 = (-100 - 300) / 32
= -400 / 32
= -12.5.

Since time cannot be negative in this context, we discard t2 = -12.5.

Therefore, the time it takes for the coyote to reach the desert floor is approximately 6.25 seconds (t1 = 6.25).

I hope this explanation helps you understand how to approach and solve this word problem!