Finding the slope of a line that cuts an area in half A region $A$ in the first quadrant is bounded by $y=x^2$, $y=25$ and the $y$-axis. Find the value of $m$ with the property that the line $y = mx$ divides $A$ into two regions with the same area.
 A: First you can calculate the area of the region A, therefore note that 5^2=25.
Hence
$$
Area_A = \int_0^5 25- x^2 \; dx = \frac{250}3.
$$
Now consider the line $g(x)=5\cdot x$. This line goes through $(0/0)$ as well as $(5/25)$.
The area between $g$ the $y$-axis and $y=25$ is a triangle with area
$$
Area_{Triangle} = \frac{25 \cdot 5}2 =\frac{125}2> \frac{Area_A}2 =\frac{125}3.
$$
This means that the line we are looking for, $h(x)=m\cdot x$ has to satisfy m>5. Taking this into account, the are $B$ bounded by  $h$ the $y$-axis and $y=25$ is again a triangle and therefore
$$
Area_B=\frac{25\cdot \frac{25}m}2 =\frac{Area_A}2= \frac{125}3.
$$
We see that $m = \frac{15}2$.
A: First, let's calculate the area of your bounded region. Let's draw a graph:

Consider your bounded area. Since $5^2=25$, It is the blue rectangle's area minus the area under $f(x)=x^2$ from $0$ to $5$. We can represent this as
$$5(25)-\int_{0}^{5}x^2\text{ }dx\text{.}$$
This is simply equal to
$$125-\frac{125}{3}=\frac{250}{3}\text{.}$$
If we consider the line $y=5x$ (the purple line in the following diagram), then this splits the blue rectangle into two equal parts, as shown.

The purple line splits the region into two regions of unequal size, and the top-left region has greater area than the bottom right one. Clearly, the line $y=mx$ must intersect the green line $y=25$ in the domain $(0,5)$.

We set up an equation in terms of $m$. The area of the orange region can be computed using the formula for the area of a triangle. If you work it out, you get
$$\frac{625}{4m}\text{.}$$
The red region is the area of the blue rectangle minus the area of the orange region and the area under $f(x)=x^2$ from $0$ to $5$. Thus, it is
$$\frac{125}{3}-\frac{625}{4m}\text{.}$$
We set these equal, so we must solve the equation
$$\frac{625}{4m}=\frac{125}{3}-\frac{625}{4m}\text{.}$$
Can you continue from here?
