Find the area of a triangle bounded by the y axis, the line $ f(x) = 4 - \frac{2}{5}x $, and the line...

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Find the area of a triangle bounded by the y axis, the line $ f(x) = 4 - \frac{2}{5}x $, and the line perpendicular to $ f(x) $ that passes through the origin. Area =

SnapAndSolve AI · Accepted Answer

To find the area of the triangle, we need to determine the points of intersection of the lines and the y-axis.

The line $ f(x) = 4 - \frac{2}{5}x $ intersects the y-axis at $ (0, 4) $.
The slope of the line $ f(x) = 4 - \frac{2}{5}x $ is $ -\frac{2}{5} $. The line perpendicular to this will have a slope of $ \frac{5}{2} $ (negative reciprocal).
The perpendicular line passing through the origin is $ y = \frac{5}{2}x $.
Find the intersection of $ y = \frac{5}{2}x $ and $ f(x) = 4 - \frac{2}{5}x $:

Set $ \frac{5}{2}x = 4 - \frac{2}{5}x $.

Multiply through by 10 to clear fractions: $ 25x = 40 - 4x $.

Solve for $ x $:

$ 25x + 4x = 40 $

$ 29x = 40 $

$ x = \frac{40}{29} $

Substitute back to find $ y $:

$ y = \frac{5}{2} \times \frac{40}{29} = \frac{100}{29} $

The vertices of the triangle are $ (0, 0) $, $ (0, 4) $, and $ \left( \frac{40}{29}, \frac{100}{29} \right) $.
The base of the triangle is the y-axis segment from $ (0, 0) $ to $ (0, 4) $, which is 4 units.
The height of the triangle is the x-coordinate of the intersection point $ \frac{40}{29} $.

Area = $ \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times \frac{40}{29} = \frac{80}{29} $.

Area = $ \frac{80}{29} $

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Find the area of a triangle bounded by the y axis, the line \( f(x) = 4 - \frac{2}{5}x \), and the line...

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