Find the equation of the line through the point (3, 5) that cuts off the least area from the first quadrant?

Wendy Larsen

Wendy Larsen

Answered question

2023-01-04

Find the equation of the line through the point (3, 5) that cuts off the least area from the first quadrant?

Answer & Explanation

Rihanna Gamble

Rihanna Gamble

Beginner2023-01-05Added 12 answers

A line through (0, 2) and (1, 1) would have the smallest area if the given point was (1, 1). (2, 0). Stretch this answer both vertically and horizontally (by a factor of 3) after that (by a factor of 5).
Let's verify using a little calculus and algebra.
Suppose the line passes through (t, 0) and (3, 5), where t>3.
Then the y intercept is at (0,5tt3)
Consequently, the triangle's area is:
12t5tt3=5t22(t3)
Therefore:
ddt5t22(t3)=5tt35t22(t3)2
ddt5t22(t3)=5t2(t3)2(2(t3)t)
ddt5t22(t3)=5t2(t3)2(t6)
Since we require t>3, the zero of the derivative that we see is when t=6, confirming the geometric solution proposed above.
The equation for this line is as follows:
5x+3y30=0

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