Writing in mathematics

Question

Directions: A quadratic function is a type of non-linear function that models certain situations. As an object falls, its speed continues to increase, so it’s height above the ground decreases at a faster and faster rate. One can model the object’s height with the function h=-16t^2 + c. The height H is given in feet above the ground after the time t is given in seconds, and the objects initial height c is given in feet. Using the text editor, write a brief essay in which you describe the difference between a linear function might go together, and what the graph might look like. Also, describe the plan or steps to solving the problem below to help you support your assertions:

A pinecone drops from a tree branch that is 20 feet above the ground. The function h=-16t^2 + 20 is used. If the height h of the pinecone is in feet after t seconds, at about what time does the pine cone hit the ground?

Answer 1

Hi Joanna! Great question: your explanation of the quadratic function and argument for why it is a good model of falling objects is solid. Let's answer your question.

You ask about why we use a quadratic function instead of a linear function. You are spot-on when you note that as objects fall, they pick up speed. A linear function would imply that the object falls at a constant rate at all times, which is not the behavior we want. A quadratic function is one way to model a change in speed, or acceleration. There are also interesting reasons from physics that support the idea that quadratic models are good at describing motion under gravity!

Now on to solving: you want to find the time t at which the pinecone hits the ground, after falling from an initial height of 20 feet. We can do this by solving for when H(t) = 0, using our equation for H(t):

H(t) = 0 = -16 t^2 + 20

There are lots of ways to solve a quadratic equation. Here, we can move (-16t^2) to the other side to get

16t^2 = 20

now we can divide by 16 and reduce our fraction to get

t^2 = 20/16 = 5/4

and now we can take the square root to find

t = square root (5)/square root(4) = square root(5)/2 which is approximately 1.12. This tells us that 1.12 seconds after being released, the pinecone will hit the ground.

Note that there is a second solution: t= -1.12 also satisfies t^2 = 5/4. In order to decide which solution is appropriate, we need to look at our application. We're interested in finding the number of seconds after which a pinecone will hit the ground. A negative number of seconds would mean that the pinecone hits the ground before it's been released: clearly that answer doesn't fit our application, so we can discard it.

A more general way to solve quadratic equations involves a formula for the solution, which you may have heard of:

To solve the equation ax^2 + bx + c = 0, we use the formula

x = -(b) +/- square root (b^2 - 4 ac)

This formula lets us find values of x that make the equation equal to 0.

Answer 2

The function is h =-16t² + 20. You are looking for the time (t) when the height (h) = 0 so:

h =-16t² + 20

0 =-16t² + 20

-20 = -16t²

-20/-16 = t²

1.25 = t²

t = √1.25 = 1.118 s

The time is about 1.1 seconds