The curve y = 11x - 24 - x^2 cuts the x-axis at points A and B, and PN is the greatest positive value of the y coordinate. Show that 2 PN • AB equals three times the area bounded by that portion of the curve which lies in the first quadrant.
Kinda confused on how to do this
well, let's find the vertex and intercepts.
x^2 - 11 x = -y - 24
x^2 - 11 x + 121/4 = -y - 24 + 30.25
(x-11/2)^2 = -(y-6.25)
so PN = 6.25
now what are A and B
when y = 0
(x-11/2)^2 = 6.25
x-5.5 = +/- 2.5
x = 3 or 8
A is at 3 and B is at 8
so 2 PN * AB = 62.5
divide by three = 20.83333.....
now the integral
from 3 to 8
of (-x^2 +11 x - 24)dx
-x^3/3 + 11x^2/2 - 24 x
at 8
-10 2/3
at 3
-31.5
20.8333... Whew !
To show that 2PN • AB equals three times the area bounded by the curve in the first quadrant, we need to break down the problem into smaller steps. Let's go through these steps one by one:
Step 1: Find the x-coordinates of points A and B.
To find the x-intercepts of the curve, set y equal to zero and solve the equation:
0 = 11x - 24 - x^2
Simplifying, we get:
x^2 - 11x + 24 = 0
This quadratic equation can be factored as:
(x - 3)(x - 8) = 0
Setting each factor equal to zero, we find the x-coordinates of A and B:
x = 3 and x = 8
Step 2: Determine the y-coordinates of A and B.
To find the y-coordinates of A and B, substitute the x-values into the equation of the curve:
For point A:
y = 11(3) - 24 - (3^2)
y = 33 - 24 - 9
y = 0
Therefore, A has coordinates (3, 0).
For point B:
y = 11(8) - 24 - (8^2)
y = 88 - 24 - 64
y = 0
Thus, B also has coordinates (8, 0).
Step 3: Calculate the greatest positive value of the y-coordinate, PN.
To find the greatest positive value of the y-coordinate, we need to determine the highest point on the curve. Since the curve is a downward-facing parabola, the vertex will give us this point. The x-coordinate of the vertex can be found using the formula x = -b / (2a) for a quadratic equation in the form ax^2 + bx + c:
In this equation, a = -1, b = 11, and c = -24.
Substituting these values:
x = -11 / (2 * -1)
x = 11 / 2
Now, substitute this x-value into the equation of the curve to find the y-coordinate:
y = 11(11/2) - 24 - ((11/2)^2)
y = 121/2 - 24 - 121/4
y = 121/2 - 96/4 - 121/4
y = 121/2 - 217/4
y = -15/4
So, the greatest positive y-coordinate is PN = -(-15/4) = 15/4.
Step 4: Calculate the length AB.
The length AB can be found by taking the absolute difference between the x-coordinates of A and B:
AB = |8 - 3|
AB = 5
Step 5: Calculate the area bounded by the curve in the first quadrant.
To calculate the area bounded by the curve in the first quadrant, we integrate the curve from x = 0 to x = 8, and then take the absolute value of the result:
Area = |∫[0, 8] (11x - 24 - x^2) dx|
Integrating the equation, we get:
Area = |[11x^2/2 - 24x - (x^3/3)] from 0 to 8|
Area = |[11(8^2)/2 - 24(8) - (8^3/3)] - [11(0^2)/2 - 24(0) - (0^3/3)]|
Area = |[352 - 192 - 512/3] - [0]|
Area = |[-352/3]|
Area = 352/3
Step 6: Show that 2PN • AB equals three times the area.
Now, we can calculate the left-hand side (LHS) and right-hand side (RHS) to compare:
LHS = 2PN • AB = 2 * (15/4) * 5 = 150/4 = 75/2
RHS = 3 * Area = 3 * (352/3) = 3 * 352/3 = 352
Since LHS = RHS, it is confirmed that 2PN • AB equals three times the area bounded by the portion of the curve in the first quadrant.