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Step 1: Identify the Enclosed Region
The region is bounded by the lines:
• x = 0
• y = 0
• x = 3/2
and the curve
y = 1 + 4x - x^2.
The line y = mx is supposed to bisect the area under the given curve within these limits.
Step 2: Find the Total Area Under the Curve
The total area under the curve y = 1 + 4x - x^2 from x = 0 to x = 3/2 is calculated by the definite integral:
\int_{0}^{3/2} \bigl(1 + 4x - x^2\bigr)\,dx.
Compute this integral:
\int \bigl(1 + 4x - x^2\bigr)\,dx
= x + 2x^2 - \frac{x^3}{3}.
Evaluating from 0 to 3/2 :
\left[x + 2x^2 - \frac{x^3}{3}\right]_{0}^{3/2}
= \left(\frac{3}{2}\right) + 2\left(\frac{3}{2}\right)^2
- \frac{\left(\frac{3}{2}\right)^3}{3}.
Simplify each term:
\frac{3}{2} = 1.5
2\left(\frac{3}{2}\right)^2 = 2 \times \frac{9}{4} = \frac{9}{2}
\left(\frac{3}{2}\right)^3 = \frac{27}{8} , and dividing by 3 gives \frac{27}{8} \times \frac{1}{3} = \frac{9}{8}
So the integral becomes:
\frac{3}{2} + \frac{9}{2} - \frac{9}{8}
= \frac{3 + 9}{2} - \frac{9}{8}
= \frac{12}{2} - \frac{9}{8}
= 6 - \frac{9}{8}
= \frac{48 - 9}{8}
= \frac{39}{8}.
Thus, the total area under the curve is \displaystyle \frac{39}{8} .
Step 3: Express the Area Under the Line y = mx
The area under the line y = mx from x=0 to x=3/2 is:
\int_{0}^{3/2} mx \,dx.
Evaluate this integral:
\int_{0}^{3/2} mx \,dx
= m \int_{0}^{3/2} x \,dx
= m \left[\frac{x^2}{2}\right]_{0}^{3/2}
= m \left(\frac{(3/2)^2}{2}\right)
= m \left(\frac{9/4}{2}\right)
= m \cdot \frac{9}{8}
= \frac{9m}{8}.
Step 4: Set Up the Bisecting Condition
The line y = mx divides the total area into two equal parts. Therefore, the area under y = mx from 0 to 3/2 must equal half of the total area under the curve. Hence,
\frac{9m}{8} = \frac{1}{2}\left(\frac{39}{8}\right).
The right side is half of \displaystyle \frac{39}{8} , which is \displaystyle \frac{39}{16} .
Step 5: Solve for m and Find 12m
Setting up the equation:
\frac{9m}{8} = \frac{39}{16}.
Multiply both sides by 16 to clear denominators:
9m \times 2 = 39 \quad \Longrightarrow \quad 18m = 39
\quad \Longrightarrow \quad m = \frac{39}{18} = \frac{13}{6}.
Finally, we need 12\,m :
12\,m = 12 \times \frac{13}{6} = 2 \times 13 = 26.
Answer
The value of 12m is \boxed{26} .
