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Okay, I'm not asking how to find the solution to this problem. I already found the solution they were looking for. The thing that confuses me is that I got two different solutions using two methods that should have given me the same answer. Could someone show me what I did wrong?
Find the distance d0,2 traveled by the car between t = 0 s and t = 2 s.
This graph is given:
Equations:
For t = 0s to t = 1s: V(t) = 30t
For t = 1s to t = 2s: V(t) = 20t
The distance is the area under the curve. You should be able to solve this either with calculus or with geometry.
I used calculus first. I took the definite integral of both equations and added them together.
The definite integral from 0 to 1 of 30t is 15. (15(1)^2 - 15(0)^2 = 15)
The definite integral from 1 to 2 of 20t is 30. (10(2)^2 - 10(1)^2 = 30)
Adding them together, the total distance traveled should have been 45 m.
This answer was incorrect. So I tried using geometry.
The first interval was a triangle. The area of a triangle is (1/2)b*h.
For 0 to 1: b = 1, h = 30. (1/2)(1*30) = 15. Same as the definite integral.
The second interval was a triangle on top of a rectangle. The area of a rectangle is b*h.
For 1 to 2: b = 1, hrectangle = 30, htriangle = 20.
(1*30) + (1/2)(1*20) = 40. Not the same as the definite integral.
Adding them together, I got 55, which was the correct answer.
So, now for my question: Why aren't they the same? Did I make a mistake somewhere, or is my understanding incorrect?
Homework Statement
Find the distance d0,2 traveled by the car between t = 0 s and t = 2 s.
This graph is given:
Homework Equations
Equations:
For t = 0s to t = 1s: V(t) = 30t
For t = 1s to t = 2s: V(t) = 20t
The Attempt at a Solution
The distance is the area under the curve. You should be able to solve this either with calculus or with geometry.
I used calculus first. I took the definite integral of both equations and added them together.
The definite integral from 0 to 1 of 30t is 15. (15(1)^2 - 15(0)^2 = 15)
The definite integral from 1 to 2 of 20t is 30. (10(2)^2 - 10(1)^2 = 30)
Adding them together, the total distance traveled should have been 45 m.
This answer was incorrect. So I tried using geometry.
The first interval was a triangle. The area of a triangle is (1/2)b*h.
For 0 to 1: b = 1, h = 30. (1/2)(1*30) = 15. Same as the definite integral.
The second interval was a triangle on top of a rectangle. The area of a rectangle is b*h.
For 1 to 2: b = 1, hrectangle = 30, htriangle = 20.
(1*30) + (1/2)(1*20) = 40. Not the same as the definite integral.
Adding them together, I got 55, which was the correct answer.
So, now for my question: Why aren't they the same? Did I make a mistake somewhere, or is my understanding incorrect?