$\displaystyle\int_C\mathrm ds = \int_a^b \sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2} \,\mathrm dt$

In this case, we have

x(t) = exp(t ) + exp(-t ) ==> dx/dt = exp(t ) - exp(-t )

y(t) = 5 - 2t ==> dy/dt = -2

and [a, b] = [0, 2]. The length of the curve is then

$\displaystyle\int_0^2 \sqrt{\left(e^t-e^{-t}\right)^2+(-2)^2} \,\mathrm dt = \int_0^2 \sqrt{e^{2t}-2+e^{-2t}+4}\,\mathrm dt$

$=\displaystyle\int_0^2 \sqrt{e^{2t}+2+e^{-2t}} \,\mathrm dt$

$=\displaystyle\int_0^2\sqrt{\left(e^t+e^{-t}\right)^2} \,\mathrm dt$

$=\displaystyle\int_0^2\left(e^t+e^{-t}\right)\,\mathrm dt$

$=\left(e^t-e^{-t}\right)\bigg|_0^2 = \left(e^2-e^{-2}\right)-\left(e^0-e^{-0}\right) = \boxed{e^2-\frac1{e^2}}$

5 0

2 years ago

Write an equation for the statement: A kilogram is equal to approximately 2.2 pounds. A cinder block weighs x pounds. A brick ha

navik [9.2K]

the total mass in kilograms y of the cinder block and brick is $y=0.45x + 0.6$ .

Step-by-step explanation:

Here we have , A kilogram is equal to approximately 2.2 pounds. A cinder block weighs x pounds. A brick has a mass of 0.6 kilogram. We need to find What is the total mass in kilograms y of the cinder block and brick . Let's find out :

According to given data in question ,

⇒ $1kg = 2.2pounds$

⇒ $1(pounds) = \frac{1}{2.2}kg$

A cinder block weighs x pounds and a brick has a mass of 0.6 kilogram . Let total mass in kg of the cinder block and brick is y , i.e.

⇒ $y (kg)= x (pounds) + 0.6 kg$

⇒ $y (kg)= x (\frac{1}{2.2}kg) + 0.6 kg$

⇒ $y=0.45x + 0.6$

Therefore , the total mass in kilograms y of the cinder block and brick is $y=0.45x + 0.6$ .

7 0

3 years ago

-2x-9y=-25 -4x-9y=-23

goldfiish [28.3K]

So to start this off you would add 9y to both sides of the equation and then you regroup it which gives you −2x=−25+9y and then you regroup the terms and get -2x=9y-25 and then you dived both sides by -2 and get x=9y-25/-2 then you simplify and so on

but at the end you should get x=-1 and y =3

8 0

3 years ago