Find the number b such that the line y = b divides the region bounded by the curves y = 16x2 and y = 9 into two regions with equal area. (Round your answer to two decimal places.)
Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (4, 64), and the x-axis.
To find the number b that divides the region into two equal areas, we need to set up an equation and solve for b.
First, let's find the points of intersection between the two curves y = 16x^2 and y = 9.
Setting the two equations equal, we have:
16x^2 = 9
Dividing by 16:
x^2 = 9/16
Taking the square root of both sides:
x = ±√(9/16)
So the two x-values of intersection are x = 3/4 and x = -3/4.
To find the y-values at these points, we substitute the x-values into either of the original equations:
For the curve y = 16x^2:
y = 16 * (3/4)^2 = 9
So, one point of intersection is at (3/4, 9).
To divide the region into equal areas, we need to find another point of intersection where the area between the curves is equal.
The region between the curves can be found by taking the definite integral of the difference between the curves over the interval where they intersect.
Let's integrate the difference between the curves from x = -3/4 to x = 3/4:
A = ∫(9 - 16x^2) dx from -3/4 to 3/4
Integrating, we get:
A = [9x - (16/3)x^3] from -3/4 to 3/4
Substituting the limits of integration:
A = [9(3/4) - (16/3)(3/4)^3] - [9(-3/4) - (16/3)(-3/4)^3]
Simplifying the expression:
A = (27/4 - 27/64) - (-27/4 + 27/64)
A = 243/64
In order for the region to be divided into two equal areas, each area should be half of this value:
243/64 = 2 * (243/64) / 2
Let's set up an equation to find b:
2 * (243/64) / 2 = ∫(b - 16x^2) dx from -3/4 to b
Integrating, we get:
243/64 = [bx - (16/3)x^3] from -3/4 to b
Substituting the limits of integration and simplifying:
243/64 = (b * b - (16/3)b^3) - (9/4 + (16/3)*(3/4)^3)
Simplifying further:
243/64 = (b * b - (16/3)b^3) - (9/4 + 27/64)
Multiplying by the common denominator of 64:
243 = 64b^3 - 16b^3 - 9 * 64 - 27
Collecting like terms:
0 = 64b^3 - 16b^3 - 9 * 64 - 27 - 243
0 = 48b^3 - 9 * 64 - 270
Now we need to solve this equation for b. Since it is a cubic equation, we can use numerical methods or software to find the solution. Rounding to two decimal places, we find:
b ≈ 2.25
Therefore, the number b that divides the region into two regions with equal areas is approximately 2.25.