The cost of the product is very expensive.

The price of the product is very high. a parabola

Parabolic mirrors have a particular shape, a parabola, and are used to focus signals. This is why they are used in satellite dishes. They can also be used in listening devices to focus a weak sound signal, which can then be amplified. This same shape is used in torches and spotlights to focus the beam. To understand why it has to be a parabola shape to get this focusing behavior, we need to look at the math and geometry behind it. The equation for a parabola is y^2 = 4ax, where a controls how steep the parabola is. If x is set to at^2, then y is equal to 2at (plus or minus). This is how a parabolic mirror works: light or sound is focused onto a single point, which is why it is used in TV signals, listening devices, torches, and spotlights. The light or sound is bounced off the parabola and converges at a point a0 inside the parabola. This amplifies the signal and creates a focused beam in the case of a torch. We have to do implicit differentiation which tells us that $$\frac{dy}{dx} = \frac{4A}{2y}$$ We want to know the gradient at point $P$: $$\frac{dy}{dx} = \frac{4A}{2y} = \frac{4A}{2 \cdot 2A} = \frac{1}{t}$$ The gradient at point $P$ is $$\frac{dy}{dx} = \frac{1}{t}$$ We can see that the gradient gets steeper as we move in this way, and it gets flatter as we move out.

We now want to work out the direction vector between the line and the focus, so the vector from $P$ to the focus is $$l = \begin{bmatrix} a^2 - a \ 2A \end{bmatrix}$$ The tangent vector is $$t = \begin{bmatrix} t \ 1 \end{bmatrix}$$ The angle between the two vectors is given by the dot product $$l \cdot t = |l| \cdot |t| \cdot \cos \theta_2$$ where $|l| = \sqrt{a^2 - a + 4A}$ and $|t| = \sqrt{t^2 + 1}$. Therefore, $$\cos \theta_2 = \frac{a^3 - at + 2At}{\sqrt{(a^2 - a + 4A)(t^2 + 1)}}$$ Cos $\theta_2$ is equal to $\frac{t}{\sqrt{t^2+1}}$, where $t$ is a vector and $\theta_2$ is the angle between $t$ and a horizontal vector $h$. We can conclude mathematically that any horizontal ray coming in and hitting a parabolic mirror will always go to the central point, because the angle coming in is the same as the angle going out. This episode of Number Five was brought to you by Jane Street, a research-based trading firm with offices all across the globe. They have fantastic programs and current opportunities for women, transgender and gender expansive people, such as their Insight and Wise programs. These include lectures, games, mock trading sessions and more. Visit their website to learn more about their jobs and internships, and to apply for one of their programs.