Area functions Let A(x)

Question

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

c. Find a formula for A(x)

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let capital C of X be the total cost in dollars of electricity used by a factory from time T equal to 02 T equal to X hours, where the cost rate is given by lowercase CFT, which is equal to 0.5 T plus 3 when T is less than or equal to 4 and 5 when T is greater than 4, we're asked to find the formula or capital C of X. Below the question we're given a graph of our function of lowercase cfT, and it consists of two linear segments. The first segment occurs when T is less than equal to 4, and that line has a slope of 0.5 and intersects the Y axis at Y equal to 3. For the second segment is a horizontal line segment located at Y equal to 5, and it occurs when T is greater than 4. Now, as to find the formula of capital C of X, which is the total cost in dollars of the electricity used by the factory from time T equal to 0 to T equal to X. And that's going to be equal to the area underneath of our cost rate function, lowercase cFT. So, we're gonna have to break up this problem into two different parts. The first part we're going to look at is when 0 is less than or equal to X, which is less than or equal to 4. And in this case, we're gonna be looking at a value for X, that is between T equal to 0 and T equal to 4. And when we draw that on our graph, we see that we're gonna be looking at the area of a trapezoid, which is going to be the area underneath of our curve, uh lowercase CFT. So that means that capital D of X is going to be equal to the area of a trapezoid. So recall your formula for the area of a trapezoid, that's gonna be 1/2, multiplying the quantity of A plus B in quantity multiplied by H. Where A and B are the lengths of the two parallel sides of a trapezoid, and H is their separation distance. So, using that information, we can see from a graph that H is going to be equal to X, since they're separated by a distance of X. The value of A is going to be equal to the Y value on our graph, which is just the value of CFT lowercase CFT. And so that is going to be occurring at T equal to 0, or A, since we'll set A to be the left side of our trapezoid, that means that A is going to be equal to, and here we'll use our function of lowercase CFT, which is going to be 0.5, multiplied by T, which in this case is 0, plus 3. And this evaluates to 3. So A is equal to 3. We'll do the same thing for the other side, which is for side B, which is the right side. And so this is going to be equal to 0.5, and this is located at a T equal to X value, so we'll have 0.5 multiplied by X plus 3. So now we can substitute these quantities back into our expression for capital C of X. We'll get capital C of X is equal to 1/2, multiplying the quantity of A, which is 3, plus B, which is 0.5 X plus 3 in quantity multiplied by H, which is X. So now we need to simplify this expression. So this is going to be equal to, we'll multiply the X by 1/2, so we'll get X divided by 2. That's gonna be multiplying in the quantity of 0.5 X. Plus 3 + 3, which is 6. And now we'll distribute our X divided by 2, and so this is going to be equal to 0.25 multiplied by X2, plus 3 multiplied by X. So, this is the first part of our function of capital C of X completed. So now we need to look at the other case, which is when X is greater than 4. So, in this case, we're gonna have a value of X. Which is greater than 4. So we're gonna place a value of T equal to X, where that value of X is greater than 4. And if we look now at our graph and try to calculate the area underneath that curve, we see that we need to have two different areas. We'll still have our trapezoid. Area that occurs between T equal to 0 and T equal to 4, but then we will have the area of a rectangle going from T equal to 4 to T equal to X. So we're gonna have to add these two areas together. So that means that our function, capital C of X, in this case, is going to be equal to the area of the trapezoid going from T equal to 0 to T equal to 4, but that is capital C of 4, since the previous function that we calculated for capital C of X is calculating the area of a trapezoid. And so, if we set X equal to 4, that's going to give us the area of that entire trapezoid between T equal to 0 and T equal to 4. That means that all we need to add to this now is the area of that rectangle. So, we'll have to add to this. Our area of the rectangle, which is going to be its height, which is 5, multiplied by its length, and that's gonna be X minus 4. Since we start at T equal to 4 and go to T equal to X, that distance is just gonna be X minus 4. So, now we need to do is calculate our value for C of 4. Using our our function for C of X, which is between uh occurs when 0 is less than equal to X, which is less than equal to 4. So for C of 4, this is going to be equal to 0.25 multiplied by 4 squared, plus 3 multiplied by 4. And when we evaluate this expression, this is going to be equal to 16. So, we'll substitute that value back into our expression for capital CF X. So we'll have capital C of X. is equal to 16. Last, and here we'll distribute our 5, so we'll have 5 X. -20, and simplifying this, this is going to be equal to 5 X. -4. So now we have our function for C of X, for our two conditions when 0 is less than or equal to X, which is less than or equal to 4, and when X is greater than 4. So we can write these as a single piece wise function. So we'll have capital C of X. is equal to 0.25 multiplied by X squared. Plus 3 X. When 0 is less than or equal to X, which is less than or equal to 4, and 5 X minus 4. When X is greater than 4. And so this is the function that our problem want us to find, and so this is the answer to our problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

c. Find a formula for A(x)

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Key Concepts

Definite Integral

Piecewise Functions

Area Under a Curve

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