# calculus

posted by .

The base of a triangle is on the x-axis, one side lies along the line y=3x, and the third side passes through the point (1,1). What is the slope of the third side if the area of the triangle is to be a minimum.

• calculus -

Interesting problem. Check to make sure I get this right.

So, we have a triangle with base k, and vertex where the line through (1,1) and (0,k) intersects y=3x.

The line through (1,1) and (0,k) is
(y-1)/(x-1) = -1/(k-1)
y = (x-1)/(1-k) + 1

So, that line intersects y=3x when

3x = (x-1)/(1-k) + 1
x = k/(3k-2)
y = 3k/(3k-2)

The area a of the triangle is thus

a = 1/2 k * 3k/(3k-2) = 3k^2/(6k-4)

da/dk = (6k(6k-4) - 3k^2 * 6)/(6k-4)^2
= 6k(3k-4)/(6k-4)^2

da/dk = 0 when k=0 or k = 4/3
Thus, the vertex of the triangle is where the line through (1,1) and (4/3,0) intersects y=3x

(y-1)/(x-1) = 1/(-1/3)
y = -3x + 4

That line has slope -3

. . .

-3x + 4 = 3x
6x = 4
x = 2/3
y=2

So, the triangle has base k=4/3, height h=2, area a=4/3

## Similar Questions

1. ### A Geometry ?

Given that two sides of a triangle are 20 and 66. Find the range of possible measures for the third side, s. a. 20<s<66 b. 46<s<86 c. 20<s<86 d.46<s<66 If you could tell me how to do this problem, i could do …
2. ### Geomentry

What is the converse of the theorem statement: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If two sides of a triangle are divided in the same proportion, …
3. ### IRSC

the sum of the lengths of any two sides of a triangle must be greater than the third side. If a triangle has one side that 17 inches and a second side that is 1 inch less than twice the thrid side, what are the possible lengths for …
4. ### math

the sum of the lenths of any two sides of a triangle must be greater than the third side. if a triangle has one side that is 17 cm and a second side that is 4cm less than twice the third side, what are the possible lenths for the second …
5. ### math

the perimeter of a triangle is 79.75 inches. the second side of the triangle is half as big as the first side. the third side is 7.25 in greater than the first side what is the length of the longest side of the triangle?
6. ### Math

In triangle ABC, D bisects side BC, G bisects side AB, and the points E and F trisect side AC. What is the area of the shaded polygon, if the area of ABC is 108?
7. ### Math

In triangle ABC, D bisects side BC, G bisects side AB, and the points E and F trisect side AC. What is the area of the shaded polygon, if the area of ABC is 108?