. During a volleyball game, Christine dove to save a ball. The function h(t)=-4.9t^2+8.5t approximates the height of the ball above the ground in metres, t seconds after it is hit.
a. If the net is 2.23 m tall, the function d(t)=I-4.9t^2+8.5t-2.23I
represents the distance between the bottom of the ball and the top
of the net after t seconds. Explain this function.
d(t) = |h(t)-2.23|
recall that |x-y| is the distance between x and y
Well, well, well, looks like Christine is quite the volleyball player! Good for her! Now, let's break down this function.
First, we start with the given function h(t) = -4.9t^2 + 8.5t, which represents the height of the ball above the ground. No surprises there.
However, we're interested in the distance between the bottom of the ball and the top of the net after t seconds. So, we subtract the net height (2.23 m) from the ball's height.
Now, you might be wondering why we have the absolute value symbols (I...I) around the expression. Well, that's there to make sure we don't end up with any negative distances. We're measuring the distance, not the depth of despair!
So, the function d(t) = I -4.9t^2 + 8.5t - 2.23 I gives us the distance between the bottom of the ball and the top of the net at any given time.
Remember, in volleyball, the aim is to clear that net! No tangled limbs or tangled ball net situations here, please!
The function d(t) represents the distance between the bottom of the ball and the top of the net after t seconds. It is derived by subtracting the height of the net (2.23 m) from the height of the ball above the ground (h(t) = -4.9t^2 + 8.5t).
By subtracting the height of the net from the height of the ball, we get the distance above the net. This function allows us to track the specific distance between the bottom of the ball and the top of the net at any given time during the volleyball game.
To understand the function d(t) = |-4.9t^2 + 8.5t - 2.23|, we need to break it down into its components.
First, let's analyze -4.9t^2 + 8.5t - 2.23. This represents the height of the ball above the ground at a given time t, which is given by the function h(t) = -4.9t^2 + 8.5t.
Now, let's focus on d(t) = |-4.9t^2 + 8.5t - 2.23|. The absolute value notation (| |) ensures that the distance between the bottom of the ball and the top of the net is positive. It represents the vertical distance between the height of the ball and the top of the net.
To calculate this distance, we subtract the height of the net (2.23 m) from the height of the ball at a given time t. Since the net is fixed at 2.23 m, we subtract it from the height of the ball, which is given by the function h(t).
The function d(t) = |-4.9t^2 + 8.5t - 2.23| gives us the distance between the bottom of the ball and the top of the net after t seconds. It tells us how close or far the ball is from clearing the net at a particular time during the game.