Mathematics: A rope is 1012 meters long. A piece of rope 458 long is cut off from the total length of the rope. What is the remaining length?

A rope is 1012 meters long. A piece of rope 458

gmedisa08

09.03.2022

Answer with Step-by-step explanation:

We are given that

T(x,y,z)=x^2+y^2+z^2

$\sigma(t)=(cost,sint,t)$

T(t) be the temperature at time t.

a.Substitute the value of x,y and z

Where x=cost,y=sint,z=t

T(t)=cos^2t+sin^2t+t^2

We know that

sin^2x+cos^2x=1

By using the formula

T(t)=1+t^2

Now, differentiate w.r.t t

T'(t)=2t

b. $t=\frac{\pi}{6}+0.01$

We have to find the approximate value of the temperature at given value of t.

Substitute $x=\frac{\pi}{6}$

$T(\frac{\pi}{6})=1+(\frac{\pi}{6})^2=1+\frac{\pi^2}{36}$

$T'(\frac{\pi}{6})=2\times\frac{\pi}{6}=\frac{\pi}{3}$

Linear approximation formula:

$f(x)\approx L(x)=f(a)+f'(a)(x-a)$

Where $a=\frac{\pi}{6}$

and $t=\frac{\pi}{6}+0.01$

Substitute the values

The approximate value of the temperature= $T(\frac{\pi}{6})+T'(\frac{\pi}{6})(\frac{\pi}{6}+0.01-\frac{\pi}{6})$

The approximate value of the temperature= $1+\frac{\pi^2}{36}+\frac{\pi}{3}(0.01)$

The approximate value of the temperature= $1+\frac{(3.14)^2}{36}+\frac{3.14}{3}\times 0.01=1.284$

Where $\pi=3.14$

0,0(0 оценок)