What is the correct answer?

Question

A ball is thrown from a height of 5 feet. The ball will travel upward at a rate of 12 meters/second Gravity will pull the ball downward which will slow its rate about 4 meters/second2. With this information the equation below can be written to help model the path of the ball during flight. 

–4t²+12t+5

I.​ The maximum height the ball will reach is 14 feet. II.​The ball will begin to descend toward the ground in 3 seconds. III.​The ball will hit the ground 14 feet from the initial point of the toss. IV.​The maximum distance the ball will reach is between 3 and 3.5 feet. 

Which of the statements above is accurate? 

A. I and II

B. II and III

C. III and IV

D. I and IV 

Answer 1

I actually think part IV is close to correct because you can solve the equation of f(x) set equal to -5 because you can look at the -5 as the distance below from which the rocket was launched. So your quadratic becomes -4x^2+12x+5 = -5 or -4x^2+12x+10. I used the quadratic formula and the answer came out to be 3.7. The other answer was irrelevant because it was negative. So this proves that part IV incorrect as it said it is somewhere between3 and 3.5. 3.7 is close but no cigar.

Answer 2

PART 1

We can check the maximum of this function by finding the vertex of the quadratic function.

Since the function is in standard form, we know the vertex will be at

(-b/2a , f(-b/2a)

The values of a and b are a = -4 and b = 12

So, we have

(-12/-8, f(-12/-8)) = (12/8, f(12/8)) = (3/2, f(3/2))

To find f(3/2) we just substitute t with 3/2.

Thus, we have

f(3/2) = -4(3/2)² + 12(3/2) + 5 = 14

Therefore, I is true.

PART 2

Let's refer back to Part 1.

Notice the value for t was 3/2 at the vertex.

This means it takes the ball 3/2 seconds (or 1.5 seconds) to reach it's maximum.

So, it will take the same amount of time for the ball to come down.

Thus, the ball will hit the ground after 3 seconds from the throw.

Therefore, II is true.

PART 3

Since the function given can only be used to find the height and not the distance traveled, III is false.

PART 4

Again, the function given can only be used to find the height and not the distance.

Therefore, IV is false.

In conclusion, only I and II are true.

*Please feel free to ask me any question for clarification.

Answer 3

So, all this problem takes is a bit of applied mathematics. BUT, 1st I am guessing everything is in either feet or meter, prolly feet from what I see! Otherwise our equation is incoherent.

The output of the equation equals the height. When you set the equation to zero, which means a height of zero, when the object has hit the ground, you find the total time the object has been in the air.

f(h) = -4t² + 12t +5 = 0

Use the quadratic to find the zeros (the values for t), one of the values you have to discard.

Actually, when f(h) = 0 = -4t² + 12t +5 = (t - (3/2 - √(14)/2))((t + 3/2 + √(14)/2))

One of the values of t will be negative and unless you are in an alternate universe time cannot be negative. So, you can cross that one off.

*TRICK* The trick to find the max height is that you have to divide the time by 2. That's because the time for the object to go up is the same time it takes to go down.

So, when you input the t/2 you get max height, which I got about 13.86 ft.

The distance questions aren't even relevant because only height is involved.