Use calculus to find the area A of the triangle with the given vertices.
(0, 0), (3, 1), (1, 8)
A = ????
the line through (3,1) and (1,8) is 7x+2y = 23
the line through (0,0) and (1,8) is y = 8x
the line through (0,0) and (3,1) is y = 1/3 x
So, the area is
∫[0,1] (23-7x)/2-x/3 dx + ∫[1,3] 8x-x/3 dx
no sweat, eh?
See the lines at
http://www.wolframalpha.com/input/?i=plot+7x%2B2y%3D23%2Cy%3D8x%2Cy%3D1%2F3+x+for+x+%3D+-2..4
Ah, the area of a triangle! Is there an easier way to find it? Maybe we can use pick-up sticks or count the number of cheerios we can fit inside. Either way, let's crunch some numbers!
To calculate the area of a triangle using calculus, we'll use the formula:
A = 1/2 * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|
Plug in the values for the vertices:
A = 1/2 * |(0(1 - 8) + 3(8 - 0) + 1(0 - 1))|
Now let's simplify:
A = 1/2 * |(0 - 0 + 1)|
A = 1/2 * 1
A = 1/2
So, the area of the triangle is 1/2 of whatever unit you're using to measure. Keep in mind that the answer is always positive, regardless of orientation.
To find the area of a triangle using calculus, we can use the formula:
A = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Given the vertices (0, 0), (3, 1), and (1, 8), we can substitute the coordinates into the formula. Let's calculate it step-by-step:
x₁ = 0, y₁ = 0
x₂ = 3, y₂ = 1
x₃ = 1, y₃ = 8
A = 1/2 * |0(1 - 8) + 3(8 - 0) + 1(0 - 1)|
Simplifying the expression inside the absolute value:
A = 1/2 * |-7 + 24 + 0|
A = 1/2 * |17|
Since the absolute value of 17 is 17, we have:
A = 1/2 * 17
A = 17/2
A = 8.5
Therefore, the area of the triangle with vertices (0, 0), (3, 1), and (1, 8) is 8.5 square units.
To find the area of a triangle using calculus, we can use the formula:
A = 1/2 * |(x1 * (y2 - y3)) + (x2 * (y3 - y1)) + (x3 * (y1 - y2))|
First, let's assign the coordinates of the vertices to variables:
(x1, y1) = (0, 0)
(x2, y2) = (3, 1)
(x3, y3) = (1, 8)
Substituting these values into the formula, we have:
A = 1/2 * |(0 * (1 - 8)) + (3 * (8 - 0)) + (1 * (0 - 1))|
Simplifying the expression inside the absolute value, we get:
A = 1/2 * |(0 * (-7)) + (3 * 8) + (1 * (-1))|
= 1/2 * |0 + 24 - 1|
= 1/2 * |23|
Since the expression inside the absolute value is positive, we can remove the absolute value sign:
A = 1/2 * 23
A = 11.5
Therefore, the area of the triangle with the given vertices is 11.5 square units.