Question

section 2.3
score: 6/9 answered: 6/9
question 7
find the area of a triangle bounded by the y axis, the line f(x)=7 - \frac{1}{6}x, and the line perpendicular to f(x) that passes through the origin.
area =
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Explanation:

Step1: Find the slope of the perpendicular line

The slope of $f(x)=7 - \frac{1}{6}x$ is $m_1=-\frac{1}{6}$. The slope of a line perpendicular to it, $m_2$, satisfies $m_1m_2=- 1$. So $m_2 = 6$. Since the perpendicular line passes through the origin $(0,0)$, its equation is $y = 6x$.

Step2: Find the intersection point of $y = 6x$ and $y=7-\frac{1}{6}x$

Set $6x=7-\frac{1}{6}x$. Add $\frac{1}{6}x$ to both sides: $6x+\frac{1}{6}x=7$. Combine like - terms: $\frac{36x + x}{6}=7$, or $\frac{37x}{6}=7$. Solve for $x$: $x=\frac{42}{37}$. Then $y = 6x=\frac{252}{37}$.

Step3: Find the $y$-intercept of $y = 7-\frac{1}{6}x$

When $x = 0$, $y=7$.

Step4: Calculate the area of the triangle

The base of the triangle is the $y$-intercept of $y = 7-\frac{1}{6}x$, which is $b = 7$. The height of the triangle is the $x$-coordinate of the intersection point of $y = 6x$ and $y=7-\frac{1}{6}x$, which is $h=\frac{42}{37}$. The area of a triangle is $A=\frac{1}{2}bh$. So $A=\frac{1}{2}\times7\times\frac{42}{37}=\frac{147}{37}$.

Answer:

$\frac{147}{37}$