A trapezoid with area 417/16 has three of its sides on the x-axis, the line x=3, and the line x=6. The fourth side is contained in the line y=mx+5/2. The value of m can be written as a/b where a and b are coprime positive integers. Find a+b.
To solve this problem, we need to find the equation of the fourth side of the trapezoid contained in the line y = mx + 5/2. Then, we can calculate the value of m and find its numerator and denominator to determine a and b.
First, let's find the length of the two parallel sides of the trapezoid. The lengths of these two sides are the differences between their respective x-values.
The length of the lower parallel side is 3 - 0 = 3 units.
The length of the upper parallel side is 6 - 0 = 6 units.
Next, let's find the height of the trapezoid, which is the distance between the x-axis and the line y = mx + 5/2. We know that the line y = mx + 5/2 crosses the x-axis at some point. To find the x-coordinate of this point, we need to substitute y = 0 into the equation:
0 = mx + 5/2
mx = -5/2
x = -5/2m
This tells us that the line crosses the x-axis at x = -5/2m. However, we also know that this line's x-intercept is included in the trapezoid's three sides on the x-axis. So, we need to find the intersection point between x = -5/2m and x = 3.
Setting these two equations equal to each other:
-5/2m = 3
Next, we solve for m:
m = -5/6
Now we have found the value of m. To determine a and b, we need to express -5/6 as a fraction in lowest terms. The numerator is -5, and the denominator is 6. Since -5 and 6 do not share any common factors other than 1, we can say that a = 5 and b = 6.
Therefore, a + b = 5 + 6 = 11.
So, the value of a + b is 11.