Question

Find the equation of the line through P=(8,7) such that the triangle bounded by this line and the axes in the first quadrant has the minimal area.

Answer 1

The area of the triangle is given by the product of 0.5 and the distances of the x and y intercepts from the origin.

• $\mathrm{The \ equation \ of \ the \ line \ is, }\displaystyle \ \underline{ y = 14 - \frac{7}{8} \cdot x}$

Reasons:

The area of the triangle, A = 0.5·x·y

Where;

x and y are the values of the line at the intercepts

The equation of the line is; (y - 7) = m·(x - 8)

At the y-intercept, x = 0, therefore;

(y - 7) = m·(0 - 8) = -8·m

At the y-intercept, y = 7 - 8·m

At the x-intercept, y = 0, therefore;

(0 - 7) = m·(0 - 8) = -8·m

At the x-intercept, -7 = m·x - 8·m

8·m - 7 = m·x

$\displaystyle x = \mathbf{8 - \frac{7}{m}}$

Therefore;

$\displaystyle A = 0.5 \times \left(7 - 8 \cdot m\right) \times \left(8 - \frac{7}{m} \right) = \mathbf{\frac{-32 \cdot m^2 +56 \cdot m - 24.5}{m}}$

When the area is minimal, we have;

$\displaystyle \frac{dA}{dm} =0 = \frac{d}{dm} \left(\frac{-32 \cdot m^2 +56 \cdot m - 24.5}{m}\right) = \mathbf{-\frac{32 \cdot m^2 - 24.5}{m^2}}$

m² × 0 = 24.5 - 32·m²

$\displaystyle m^2 = \frac{24.5}{32} = \frac{49}{64}$

$\displaystyle m = \sqrt{\frac{49}{64}} = \frac{7}{8}$

$\displaystyle m = \pm \frac{7}{8}$

The equation of the line when $\displaystyle m = + \frac{7}{8}$ is therefore;

(y - 7) = m·(x - 8)

$\displaystyle (y - 7) = \mathbf{\frac{7}{8} \cdot (x - 8)}$

$\displaystyle (y - 7) = \frac{7}{8} \cdot (x - 8) = \frac{7}{8} \cdot x - \frac{7}{8} \times 8 = \frac{7}{8} \cdot x - 7$

$\displaystyle y = \frac{7}{8} \cdot x - 7 + 7 = \frac{7}{8} \cdot x$

$\displaystyle y = \mathbf{\frac{7}{8} \cdot x}$

The x and y intercept of the above line are 0

When $\displaystyle m = - \frac{7}{8}$, we have;

$\displaystyle (y - 7) = -\frac{7}{8} \cdot (x - 8) = -\frac{7}{8} \cdot x + \frac{7}{8} \times 8 = -\frac{7}{8} \cdot x + 7$

Which gives;

$\displaystyle y = \frac{7}{8} \cdot x + 7 + 7 = \frac{7}{8} \cdot x + 14$

$\mathrm{The \ equation \ of \ the \ line \ is, }\displaystyle \ y = \mathbf{14 - \frac{7}{8} \cdot x}$

The equation of the line through P = (8, 7) such that the triangle bounded by the line and the axes in the first quadrant has minimal area is therefore;

$\displaystyle \ \underline{ y = 14 - \frac{7}{8} \cdot x}$

Learn more here:

https://brainly.com/question/4032024

https://brainly.com/question/1621381