# Maths - Modelling with Differentiation

> Source: https://ollybritton.com/notes/a-level/maths/topics/modelling-with-differentiation/ · Updated: 2024-08-25 · Tags: maths, differentiation

##### What would the differential be called for $A = \pi r^2$??
$$
\frac{dA}{dr}
$$

##### $$A = \pi r^2 \\ \frac{dA}{dr}$$ How would you describe the differential??
The rate of change of area with respect to radius.

##### Can you differentiate $V = \frac{4}{3} \pi r^3$??
$$
\frac{dV}{dr} = 4\pi r^2
$$

##### $$V = \frac{4}{3} \pi r^3 \\ \frac{dV}{dr} = 4\pi r^2$$ How could you explain "the rate of change of volume with respect to radius"??
How much additional volume you gain for a small change in the radius.

### 2021-01-29
##### ![PHOTO CUBOID XY](paste-bef81999f809b99101115bff2ea8d35a6d43b72b.jpg) This cubiod represents a tank with no top and area $54m^2$. What's the formula for the surface area??
$$
54m^2 = 2x^2 + 3xy
$$

##### ![PHOTO CUBOID XY](paste-bef81999f809b99101115bff2ea8d35a6d43b72b.jpg) What's the formula the volume of this cubiod??
$$
x^2y
$$

##### ![PHOTO CUBIOD XY](cubiod-xy.png) You have the two equations $$A = 2x^2 + 3xy = 54m^2 \\ V = x^2y$$ How would you find the actual volume of the cubiod??
Rearrange the first formula in terms of $y$ and then substitue back into the volume formula.

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