4. find the equation of the tangent line to the curve y = cos(x) at x = π.\n5. the surface area of a cube of…

Explanation:

Step1: Find the derivative of $y = \cos(x)$

The derivative of $\cos(x)$ is $y'=-\sin(x)$.

Step2: Find the slope of the tangent line at $x = \pi$

Substitute $x=\pi$ into $y'$. So $m = -\sin(\pi)=0$.

Step3: Find the y - coordinate at $x=\pi$

Substitute $x = \pi$ into $y=\cos(x)$. Then $y=\cos(\pi)=- 1$.

Step4: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$. Here $(x_1,y_1)=(\pi,-1)$ and $m = 0$. So $y-(-1)=0(x - \pi)$, which simplifies to $y=-1$.

Answer:

$y=-1$

Explanation for second question:

Step1: Differentiate the surface - area formula with respect to time $t$

Given $S = 3x^{2}$, using the chain rule $\frac{dS}{dt}=6x\frac{dx}{dt}$.

Step2: Substitute the given values of $x$ and $\frac{dx}{dt}$

We know that $\frac{dx}{dt}=0.4$ inches per second and $x = 20$ inches. Substitute these values into $\frac{dS}{dt}=6x\frac{dx}{dt}$. Then $\frac{dS}{dt}=6\times20\times0.4$.

Step3: Calculate the value of $\frac{dS}{dt}$

$\frac{dS}{dt}=6\times20\times0.4 = 48$ square inches per second.

Answer:

48 square inches per second