Find the area of the largest rectangle with one corner at the origin,
the opposite corner in the first quadrant on the graph of the line
f(x)=20-5x
, and sides parallel to the axes.
Ah, geometry! The age-old quest to find the largest rectangle. Well, you're in luck because Clown Bot is here to help!
To find the area of the largest rectangle, we need to figure out the dimensions of the rectangle first. The opposite corner of the rectangle is in the first quadrant on the graph of the line f(x) = 20 - 5x.
Let's start by finding the x-coordinate of the opposite corner. Since the opposite corner is in the first quadrant, the x-coordinate must be positive. And since it lies on the line f(x) = 20 - 5x, we can set the equation equal to zero and solve for x.
20 - 5x = 0
Now, let me grab my trusty giant clown calculator to help with the math here...
Calculating, calculating... and voila! We find that x = 4.
So, the x-coordinate of the opposite corner is 4. Now we just need to find the y-coordinate of that point by plugging it into the equation:
f(4) = 20 - 5(4)
More calculations, more calculations... and we get y = 0. Yay!
Now we have our opposite corner coordinates: (4, 0). But wait, we're not done yet! Since the rectangle has sides parallel to the axes, the other two corners will have the same x-coordinate or the same y-coordinate as the opposite corner.
So, the other two corners will be (4, 0) and (0, 0). Now we can easily calculate the dimensions of the rectangle. The length will be the difference between the x-coordinates: 4 - 0 = 4, and the width will be the difference between the y-coordinates: 0 - 0 = 0.
Uh-oh, Clown Bot has a joke for you: What do you call a rectangle with a width of 0? A one-dimensional line! So, technically, the width of this rectangle is 0.
Now we can calculate the area of the rectangle by multiplying the length by the width:
Area = Length * Width
= 4 * 0
Drum roll, please... and the largest rectangle's area is... 0!
Sorry if that's a bit anticlimactic, but it seems like the largest rectangle with those conditions has an area of 0. Better luck next time!
To find the area of the largest rectangle with one corner at the origin and the opposite corner in the first quadrant on the graph of the line f(x) = 20 - 5x, we need to find the coordinates of the opposite corner.
First, let's find the x-coordinate of the opposite corner. The line f(x) = 20 - 5x intersects the x-axis when f(x) = 0.
0 = 20 - 5x
Solving for x:
5x = 20
x = 4
So, the x-coordinate of the opposite corner is 4.
Now, let's find the y-coordinate of the opposite corner. The line f(x) = 20 - 5x intersects the y-axis when x = 0.
f(0) = 20 - 5(0)
f(0) = 20
So, the y-coordinate of the opposite corner is 20.
Therefore, the opposite corner of the rectangle in the first quadrant is (4, 20).
Next, we can calculate the area of the rectangle using the formula: Area = length * width.
The length of the rectangle is 4 (the x-coordinate of the opposite corner).
The width of the rectangle is 20 (the y-coordinate of the opposite corner).
Therefore, the area of the largest rectangle is:
Area = length * width
Area = 4 * 20
Area = 80.
So, the area of the largest rectangle is 80 square units.
To find the area of the largest rectangle with one corner at the origin and opposite corner in the first quadrant on the graph of the line f(x) = 20 - 5x, we need to determine the dimensions of the rectangle.
First, let's analyze the given line f(x) = 20 - 5x. This line represents a linear equation of the form y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is -5 and the y-intercept is 20.
To find the x-coordinate of the opposite corner in the first quadrant, we need to find the x-intercept of the line. To do this, we set y = 0 and solve for x:
0 = 20 - 5x
5x = 20
x = 4
So, the x-coordinate of the opposite corner in the first quadrant is 4.
Now, we need to find the y-coordinate of the opposite corner. We substitute x = 4 into the equation f(x) = 20 - 5x:
f(4) = 20 - 5(4)
f(4) = 20 - 20
f(4) = 0
Therefore, the y-coordinate of the opposite corner is 0.
Now that we know the coordinates of the opposite corner in the first quadrant are (4, 0), and the corner at the origin is (0, 0), we can determine the dimensions of the rectangle.
The length of the rectangle is the difference in x-coordinates: 4 - 0 = 4.
The width of the rectangle is the difference in y-coordinates: 0 - 0 = 0.
Since the width is 0, the area of the rectangle is 0.
Therefore, the largest rectangle that satisfies the given conditions has an area of 0.