# Calculating Ball Height and Time with Quadratic Formula

• MHB
In summary, the conversation discusses finding the time when a ball thrown straight upward lands and when its height is 80 feet. The best approach is to solve the given relation between height and time for t, which gives the formula t = (12 ± √(144-h)) / 4. This allows for easily plugging in any given height to find the corresponding time, and also shows that the maximum height of the ball is 144 feet, occurring at time t = 3 seconds.
A ball is thrown straight upward. Suppose that the height of the ball at time t is h = -16t^2 + 96t, where h is in feet and t is in seconds, with t = 0 corresponding to the instant the ball is first tossed.

A. How long does it take for the ball to land?

To do A, I must let h = 0 and solve for t, right?

B. At what time is the height 80 feet? Why does B have two answers?

To do B, I must let h = 80 and solve for t, right?

RTCNTC said:
A ball is thrown straight upward. Suppose that the height of the ball at time t is h = -16t^2 + 96t, where h is in feet and t is in seconds, with t = 0 corresponding to the instant the ball is first tossed.

A. How long does it take for the ball to land?

To do A, I must let h = 0 and solve for t, right?

B. At what time is the height 80 feet? Why does B have two answers?

To do B, I must let h = 80 and solve for t, right?

Yes that is what you need to do for both.

Good. I will answer both parts tonight.

Since you are asked multiple questions regarding finding the time when the ball is a certain height, what I recommend is solving the given relation between height $h$ and time $t$ for $t$, so that you then have a formula to use. We are given:

$$\displaystyle h=-16t^2+96t$$

Arrange this as:

$$\displaystyle 16t^2-96t+h=0$$

Now use the quadratic formula to obtain:

$$\displaystyle t=\frac{12\pm\sqrt{144-h}}{4}$$

Now it's just a matter of plugging in any given height to find the time when that height occurs, rather than having to solve a quadratic equation every time a new height is introduced. We can also easily see that the maximum height is 144 and occurs at time $t=3$. :D

MarkFL said:
Since you are asked multiple questions regarding finding the time when the ball is a certain height, what I recommend is solving the given relation between height $h$ and time $t$ for $t$, so that you then have a formula to use. We are given:

$$\displaystyle h=-16t^2+96t$$

Arrange this as:

$$\displaystyle 16t^2-96t+h=0$$

Now use the quadratic formula to obtain:

$$\displaystyle t=\frac{12\pm\sqrt{144-h}}{4}$$

Now it's just a matter of plugging in any given height to find the time when that height occurs, rather than having to solve a quadratic equation every time a new height is introduced. We can also easily see that the maximum height is 144 and occurs at time $t=3$. :D

Nicely done. You created an equation similar to the quadratic formula.

## What is the purpose of an application concerning a ball?

An application concerning a ball is used to study the physical properties and behavior of a ball in different situations. This can help us understand the mechanics behind sports, games, and other activities involving balls.

## What types of balls can be used in an application?

Any object that is round and can be thrown, rolled, or bounced can be considered a ball. This can include sports balls such as basketballs, soccer balls, and tennis balls, as well as everyday objects like marbles or even fruit.

## What are some potential research questions that can be explored through an application concerning a ball?

Some potential research questions could include: How does the material of a ball affect its bounce? How does air pressure inside a ball affect its flight trajectory? How does the size and weight of a ball affect its speed and distance when thrown?

## What techniques and instruments are used in an application concerning a ball?

Techniques such as measuring the bounce height, air resistance, and velocity can be used to study the behavior of a ball. Instruments such as rulers, stopwatches, and air pressure gauges can also be used to collect data and analyze the results.

## What are the real-world applications of studying balls through an application?

The knowledge gained from studying balls through an application can be applied to various real-world scenarios. For example, understanding the mechanics of a ball can help in designing better sports equipment or improving techniques in sports. It can also have practical applications in fields such as physics, engineering, and product design.

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