If the line \( y = mx \) bisects the area enclosed by the lines \( x = 0, y = 0, x = \frac{3}{2} \) and the curve \( y = 1 + 4x - x^2 \), then 12 m is equal to _________.
Show Hint
The "bisecting line" through the origin often defines a simple geometric shape like a triangle or a trapezoid. Always verify if you can use basic area formulas before performing a second integration.
Step 1: Understanding the Concept:
We first find the total area under the curve between the given vertical lines using integration. Then, we calculate the area of the region below the line \( y = mx \), which is a triangle, and set it equal to half the total area. Step 2: Detailed Explanation:
Total area \( A \):
\[ A = \int_0^{3/2} (1 + 4x - x^2) dx = [x + 2x^2 - x^3/3]_0^{3/2} \]
\[ A = \frac{3}{2} + 2(9/4) - \frac{1}{3}(27/8) = \frac{3}{2} + \frac{9}{2} - \frac{9}{8} = 6 - \frac{9}{8} = \frac{39}{8} \]
The area of the triangle formed by \( y = mx, x = 0, x = 3/2 \) is:
\[ A_{\text{tri}} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot \frac{3}{2} \cdot \left(m \cdot \frac{3}{2}\right) = \frac{9m}{8} \]
Given the line bisects the area:
\[ A_{\text{tri}} = \frac{A}{2} \implies \frac{9m}{8} = \frac{39}{16} \]
\[ m = \frac{39}{16} \cdot \frac{8}{9} = \frac{13}{2} \cdot \frac{1}{3} = \frac{13}{6} \]
Calculate 12 m:
\[ 12 \cdot \frac{13}{6} = 26 \] Step 3: Final Answer:
The value of 12 m is 26.
