R is the first quadrant region enclosed by the x-axis, the curve y = 2x + b, and the line x = b, where b > 0. Find the value of b so that the area of the region R is 288 square units.
Answer is 12
To find the value of b that makes the area of the region R equal to 288 square units, we need to find the points of intersection between the curve y = 2x + b, the line x = b, and the x-axis.
First, let's find the point of intersection between the curve and the line.
Setting y = 0 in the equation of the curve:
0 = 2x + b
Solving for x:
2x = -b
x = -b/2
Since x = b in the equation of the line, we have:
x = -b/2 = b
Setting x = b in the equation of the line:
b = b
This shows that the point of intersection between the curve and the line is (b, b).
Now, let's find the x-coordinate of the other point of intersection between the curve and the x-axis.
Setting y = 0 in the equation of the curve:
0 = 2x + b
Solving for x:
2x = -b
x = -b/2
Since we are considering the first quadrant, we only need to consider the positive value of x. Therefore, we have:
x = -b/2
Now, let's calculate the area of the region R.
The region R is a trapezoid, and its area can be calculated using the formula:
Area = (base1 + base2) * height / 2
The bases of the trapezoid are the lengths of the segments between the x-axis and the curve for the values of x from -b/2 to b.
The height of the trapezoid is b.
The length of the first segment can be calculated by substituting x = -b/2 into the equation of the curve:
y = 2x + b
y = 2(-b/2) + b
y = -b + b
y = 0
Therefore, the first base is 0.
The length of the second segment can be calculated by substituting x = b into the equation of the curve:
y = 2x + b
y = 2b + b
y = 3b
Therefore, the second base is 3b.
Using the formula for the area of a trapezoid, we have:
Area = (0 + 3b) * b / 2
288 = (3b^2) / 2
Now, let's solve this equation for b.
Multiplying both sides of the equation by 2 and dividing by 3, we get:
576 / 3 = b^2
192 = b^2
Taking the square root of both sides, we have:
b = sqrt(192)
So, the value of b that makes the area of the region R equal to 288 square units is b = sqrt(192).
Please note that sqrt refers to the square root.
To find the value of b such that the area of region R is 288 square units, we need to calculate the area of region R and solve the equation for b.
Step 1: Understanding the given information
The region R is enclosed by the x-axis, the curve y = 2x + b, and the line x = b, where b > 0. This means that region R is a triangle.
Step 2: Finding the vertices of the triangle
The vertices of the triangle are the points of intersection between the curve y = 2x + b and the line x = b.
Setting y = 2x + b equal to x = b, we can solve for the x-coordinate of the intersection point:
2x + b = b
2x = 0
x = 0
Thus, the intersection point is (0,b).
To find the y-coordinate of the intersection point, substitute the x-coordinate into the equation of the curve:
y = 2 * 0 + b
y = b
Therefore, the intersection point is (0,b).
Step 3: Calculating the area of the triangle
Since the triangle is a right-angled triangle with the x-axis as the base and the line x = b as the height, the area of the triangle is given by the formula:
Area = (1/2) * base * height
= (1/2) * (b-0) * (b-b)
= (1/2) * b * 0
= 0
Step 4: Setting up and solving the equation
We are given that the area of region R is 288 square units. Therefore, we need to solve the equation:
0 = 288
However, since this equation is not possible to solve, it means that there is no value of b that satisfies the given condition.
simple enough. You want
∫[0,b] (2x+b) dx = 288
Now just solve for b. You can check by just finding the area of the trapezoid:
b(b + 2b+b)/2 = 288