An object has dropped from a height of 600 feet. It has a height ℎ of 𝑡 in feet after 𝑡 seconds have elapsed such that ℎ of 𝑡 equals 600 minus 16𝑡 squared. Express 𝑡 as a function of height ℎ and then calculate the time taken to drop to a height of 400 feet to one decimal place.
In this question, we’ve been given a function that describes the height of an object after 𝑡 seconds. The first thing we’re told to do is express 𝑡 as a function of height ℎ. In other words, if we let ℎ be equal to 600 minus 16𝑡 squared, we’re going to change the subject of the equation. We want 𝑡 is equal to some expression in ℎ. So how do we do that? Let’s look carefully at the formula we’ve been given.
According to PEMDAS, which tells us the order in which we must perform a series of operations, we take the value of 𝑡 and then we square it. We then multiply it by negative 16 and add 600. Now, remember, this is exactly the same as multiplying it by 16 and subtracting that entire expression from 600.
To make 𝑡 the subject, we’re going to look to reverse this entire process. Whatever we do to one side of the equation, we must do to the other. So we’re going to begin by subtracting 600 from both sides. That gives us ℎ minus 600 on the left-hand side. And on the right, we’re simply left with negative 16𝑡 squared.
Next, we want to reverse the multiply by negative 16 step. So we’re going to divide both sides of our equation by negative 16. That gives us ℎ minus 600 over negative 16 equals 𝑡 squared. Now, before we do the next step, we might like to simplify this expression a little. We multiply both the numerator and the denominator of the fraction by negative one. This doesn’t change the size of the fraction. Remember, this is just creating an equivalent fraction. But what it does do is it makes the denominator of our fraction positive. So we end up with 600 minus ℎ over 16.
We then want to reverse the squaring step. So we square root both sides of our equation. And when we do, we find that 𝑡 is equal to the square root of 600 minus ℎ all over 16. Now, you might have been tempted to find both the positive and negative square root in this step. However, we’re working with time. We know time has to be a positive number. So we can disregard the negative square root.
The second part of this question asks us to calculate the time taken to drop to a height of 400 feet. So we’re going to let ℎ be equal to 400 in our function for time. And so, 𝑡 is equal to the square root of 600 minus 400 over 16. That’s the square root of 200 over 16, which is equal to 3.5355 and so on. We need to round our answer to one decimal place. Our decimal point is here. So the first decimal place is a five. The digit immediately to the right of the five is the deciding digit. And we recall that if the deciding digit is greater than or equal to five, we round the number up. And if it’s less than five, we round the number down. In our case, the number is three. That’s less than five. So we round down. And we get 𝑡 is equal to 3.5.
And so, the two solutions we’re interested in is 𝑡 is equal to the square root of 600 minus ℎ over 16 and 𝑡 equals 3.5 seconds.