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3 years ago

The area of a rectangular garden if 6045 ft2. If the length of the garden is 93 feet, what is its width?

Mathematics

2 answers:

AnnyKZ [126]3 years ago

4 0

Answer:

65 ft

Step-by-step explanation:

The area of a rectangle is

A = lw

6045 = 93*w

Divide each side by 93

6045/93 = 93w/93

65 =w

inna [77]3 years ago

4 0

Answer:

$\huge \boxed{\mathrm{65 \ feet}}$

Step-by-step explanation:

The area of a rectangle formula is given as,

$\mathrm{area = length \times width}$

The area and length are given.

$6045=93 \times w$

Solve for w.

Divide both sides by 93.

$65=w$

The width of the rectangular garden is 65 feet.

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X ^ (2) y '' - 7xy '+ 16y = 0, y1 = x ^ 4

AfilCa [17]

Given a solution $y_1(x)=x^4$ , we can attempt to find another via reduction of order of the form $y_2(x)=x^4v(x)$ . This has derivatives

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Substituting into the ODE yields

$x^2(x^4v''+8x^3v'+12x^2v)-7x(x^4v'+4x^3v)+16x^4v=0$
$x^6v''+(8x^5-7x^5)v'+(12x^4-28x^4+16x^4)v=0$
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Now letting $u(x)=v'(x)$ , so that $u'(x)=v''(x)$ , you end up with the ODE linear in $u$

$x^6u'+x^5u=0$

Assuming $x\neq0$ (which is reasonable, since $x=0$ is a singular point), you can divide through by $x^5$ and end up with

$xu'+u=(xu)'=0$

and integrating both sides with respect to $x$ gives

$xu=C_1\implies u=\dfrac{C_1}x$

Back-substitute to solve for $v$ :

$v'=\dfrac{C_1}x\implies v=C_1\ln|x|+C_2$

and again to solve for $y$ :

$y=x^4v\implies \dfrac y{x^4}=C_1\ln|x|+C_2$
$\implies y=C_1\underbrace{x^4\ln|x|}_{y_2}+C_2\underbrace{x^4}_{y_1}$

Alternatively, you can solve this ODE from scratch by employing the Euler substitution (which works because this equation is of the Cauchy-Euler type), $t=\ln x$ . You'll arrive at the same solution, but it doesn't hurt to know there's more than one way to solve this.