Show that u(x, t)=cos(a*pi*x)e^-a^2*pi^2*t is a solution of the heat equation with k=1, on any interval [0,l].

Step-by-step explanation:

We have the function

,

while the heat equation in one spacial dimension is

.

So, to solve this exercise we only need to calculate the derivatives that appears in the equation. Let us start by the derivative with respect to t:

.

In the other hand

,

then

.

Notice that,

So, the function satisfies the heat equation with on any interval [0,L].

d) 30 units^2

area of rectangle = l × b

we know,

length = 10 unit

breadth = 3 unit

applying the formula,

= l × b

= 10 × 3

= 30 unit^2

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