32 and 2.56 what is the unit rate

1. (a) Solve the differential equation (x + 1)Dy/dx= xy, = given that y = 2 when x = 0. (b) Find the area between the two curves

erastova [34]

(a) The differential equation is separable, so we separate the variables and integrate:

$(x+1)\dfrac{dy}{dx} = xy \implies \dfrac{dy}y = \dfrac x{x+1} \, dx = \left(1-\dfrac1{x+1}\right) \, dx$

$\displaystyle \frac{dy}y = \int \left(1-\frac1{x+1}\right) \, dx$

$\ln|y| = x - \ln|x+1| + C$

When x = 0, we have y = 2, so we solve for the constant C :

$\ln|2| = 0 - \ln|0 + 1| + C \implies C = \ln(2)$

Then the particular solution to the DE is

$\ln|y| = x - \ln|x+1| + \ln(2)$

We can go on to solve explicitly for y in terms of x :

$e^{\ln|y|} = e^{x - \ln|x+1| + \ln(2)} \implies \boxed{y = \dfrac{2e^x}{x+1}}$

(b) The curves y = x² and y = 2x - x² intersect for

$x^2 = 2x - x^2 \implies 2x^2 - 2x = 2x (x - 1) = 0 \implies x = 0 \text{ or } x = 1$

and the bounded region is the set

$\left\{(x,y) ~:~ 0 \le x \le 1 \text{ and } x^2 \le y \le 2x - x^2\right\}$

The area of this region is

$\displaystyle \int_0^1 ((2x-x^2)-x^2) \, dx = 2 \int_0^1 (x-x^2) \, dx = 2 \left(\frac{x^2}2 - \frac{x^3}3\right)\bigg|_0^1 = 2\left(\frac12 - \frac13\right) = \boxed{\frac13}$

7 0

2 years ago

20 people fit comfortably in a 9 feet by 10 feet area. Use this value to estimate the size of a crowd that is 7 feet deep on BOT

kari74 [83]

Answer:

389 people

Step-by-step explanation:

Step 1

we convert 1500 inches to feet

1 inch = 0.0833 feet

1500 inches = x feet

x = 1500 × 0.0833

x = 125 feet

Step 2

We create a proportion

Since it is on both sides

20/9ft × 10ft = x/7 feet × 125 feet × 2

20/90 = x/1750

Cross Multiply

90 × x = 20 × 1750

x = 20 × 1750/90

x = 35000/90

x = 388.88888889

approximately = 389 people

5 0

2 years ago

1.567 + 2.345 - 1.83(5)

g100num [7]

Answer:

-5.238

Step-by-step explanation:

Step 1: First multiply

1.567 + 2.345 - 1.83(5)

1.567 + 2.345 - 9.15

Step 2: Add the first two numbers

(1.567 + 2.345) - 9.15

3.912 - 9.15

Step 3: Add the next two numbers

3.912 - 9.15

-5.238

Answer: -5.238

6 0

2 years ago

Read 2 more answers

Zero and negative exponentswrite in simplest form without zero or negative exponents

gulaghasi [49]

We have the following rule for exponents:

$a^0=1$

then, in this case we have:

$(-17)^0=1$

7 0

9 months ago

-3x+5y=2x+3y−3x+5y=2x+3y Which ordered pair is a solution of the equation?

Lelu [443]

Answer:

beuh the answer is easy it is by by bye

Step-by-step explanation:

I'm built differently

3 0

2 years ago