Aisha and Tatia awoke in their apartment one morning to find a broken water pipe had been leaking, making a growing circular water stain on their ceiling. As one does, Aisha and Tatia began tracking the area of water damage as a function A(t), in square inches, over time t, in hours, while waiting for someone to arrive to stop the leaking pipe.
Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Area of Water Damage (in^2)
15 16 21 20 24 27 28 35 36 41 43 50 57 62 68
Q1: Since the water damage was making a circular water stain on their ceiling and knowing A = πr^2 gives the area of circle, Aisha thought a quadratic polynomial would be an appropriate function to model this event. Use the data at t = 1, 8, and 15 to create a quadratic function to model growing water damage.
-how am I supposed to make a quadratic function when the table values are not quadratic?
15 16 21 20 24 27 28 35 36 41 43 50 57 62 68
first differences:
1,5,-1,4,3,1,7,1,5,2,7,7,5,6
Doesn't look very quadratic to me; the 2nd differences will not be constant
However, given the three points, use them do determine the coefficients in
y = x^2+bx+c.
For example, (1,15) yields
a+b+c = 15
and similarly for the other two points.
To create a quadratic function to model the growing water damage, we can use the general form of a quadratic function: f(t) = a(t^2) + bt + c.
Given the data points (1, 15), (8, 35), and (15, 68), we can substitute these values into the quadratic function to obtain a system of equations:
1. 15 = a(1^2) + b(1) + c
2. 35 = a(8^2) + b(8) + c
3. 68 = a(15^2) + b(15) + c
We will solve this system of equations to find the values of a, b, and c. Here's how you can do it step-by-step:
Step 1: Substitute the values from the first data point:
15 = a(1^2) + b(1) + c
15 = a + b + c (Eq. 1)
Step 2: Substitute the values from the second data point:
35 = a(8^2) + b(8) + c
35 = 64a + 8b + c (Eq. 2)
Step 3: Substitute the values from the third data point:
68 = a(15^2) + b(15) + c
68 = 225a + 15b + c (Eq. 3)
Now we have a system of three equations with three variables (a, b, and c). Solving this system will give us the values of a, b, and c, which we can then use in the quadratic function.
I will solve this system of equations for you.
To create a quadratic function, we need to find a pattern or relationship between the time (t) and the corresponding area of water damage (A(t)) values given in the table.
In this case, since Aisha stated that a quadratic polynomial would be an appropriate function to model the event, we can assume that the area of water damage is related to the square of the time.
To do this, let's first square the time values given in the table:
Time (t) - Squared Time (t^2)
1 - 1
8 - 64
15 - 225
Now, let's examine the area of water damage (A) at these squared time values:
Squared Time (t^2) - Area of Water Damage (A)
1 - 15
64 - 35
225 - 68
By comparing the squared time values to the corresponding area of water damage values, we can see that they do not seem to fit a quadratic pattern. Instead, it looks like the relationship between time and the area of water damage is more linear.
Therefore, based on the given values and the information provided, it may not be appropriate to create a quadratic function to model the growing water damage. It might be more suitable to use a linear or different type of function instead.