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4 years ago

10

The area of Nathan's bedroom rug is 15 square feet.The longer side measures 5 feet. His living room rug is twice as long and twi

ce as wide as the bedroom rug.

1 answer:

Orlov [11]4 years ago

3 0

The one in his living room is 30 feet

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Solve the initial value problems.

slavikrds [6]

Both equations are linear, so I'll use the integrating factor method.

The first ODE

$xy' + (x+1)y = 0 \implies y' + \dfrac{x+1}x y = 0$

has integrating factor

$\exp\left(\displaystyle \int\frac{x+1}x \, dx\right) =\exp\left(x+\ln(x)\right) = xe^x$

In the original equation, multiply both sides by eˣ :

$xe^x y' + (x+1) e^x y = 0$

Observe that

d/dx [xeˣ] = eˣ + xeˣ = (x + 1) eˣ

so that the left side is the derivative of a product, namely

$\left(xe^xy\right)' = 0$

Integrate both sides with respect to x :

$\displaystyle \int \left(xe^xy\right)' \, dx = \int 0 \, dx$

$xe^xy = C$

Solve for y :

$y = \dfrac{C}{xe^x}$

Use the given initial condition to solve for C. When x = 1, y = 2, so

$2 = \dfrac{C}{1\cdot e^1} \implies C = 2e$

Then the particular solution is

$\boxed{y = \dfrac{2e}{xe^x} = \dfrac{2e^{1-x}}x}$

The second ODE

$(1+x^2)y' - 2xy = 0 \implies y' - \dfrac{2x}{1+x^2} y = 0$

has integrating factor

$\exp\left(\displaystyle \int -\frac{2x}{1+x^2} \, dx\right) = \exp\left(-\ln(1+x^2)\right) = \dfrac1{1+x^2}$

Multiply both sides of the equation by 1/(1 + x²) :

$\dfrac1{1+x^2} y' - \dfrac{2x}{(1+x^2)^2} y = 0$

and observe that

d/dx[1/(1 + x²)] = -2x/(1 + x²)²

Then

$\left(\dfrac1{1+x^2}y\right)' = 0$

$\dfrac1{1+x^2}y = C$

$y = C(1 + x^2)$

When x = 0, y = 3, so

$3 = C(1+0^2) \implies C=3$

$\implies \boxed{y = 3(1 + x^2) = 3 + 3x^2}$

7 0

2 years ago

A number that is at least 13

AveGali [126]

Answer:

13≤x

-hope this helps

8 0

4 years ago

A person runs 1/2 miles in 1/14 hour. What is the persons speed miles per hour.

Svetradugi [14.3K]

Answer:

The persons speed is 7 miles per hour.

6 0

3 years ago

Daryl hit a home run 8 out of 32 times.if he is at bat 224 times how many home runs will he hit

Nezavi [6.7K]

Answer:

56 out of 224 times

Step-by-step explanation:

8 out of 32 = 1 out of 4

56 out of 224 = 1/4

7 0

3 years ago

Read 2 more answers

solniwko [45]

Answer:

$Cost = 112.266$ cent/km

Step-by-step explanation:

Given

$Unit\ cost = R17.82/litre$

$Rate = 6.3litre/100km$

Required

The fuel cost in $cent\ per\ km$

This is calculated using:

$Cost = Unit\ cost * Rate$

$Cost = R17.82/litre * 6.3litre/100km$

$Cost = R17.82 * 6.3/100km$

$Cost = R1.12266/km$

Multiply by $100$ to get the cost in cents

$Cost = 1.12266 * 100$ cent/km

$Cost = 112.266$ cent/km

6 0

3 years ago