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professor190 [17]

2 years ago

12

a coast Guard ship patrols an area of 125 square miles. the are the ship patrols is a square. about how long is each side of the

square? Round to nearest mile.

1 answer:

Salsk061 [2.6K]2 years ago

5 0

31.25 is answer , rounded answer would be 30 miles each side.

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Use green's theorem to compute the area inside the ellipse x252+y2172=1. use the fact that the area can be written as ∬ddxdy=12∫

Pavel [41]

The area of the ellipse $E$ is given by

$\displaystyle\iint_E\mathrm dA=\iint_E\mathrm dx\,\mathrm dy$

To use Green's theorem, which says

$\displaystyle\int_{\partial E}L\,\mathrm dx+M\,\mathrm dy=\iint_E\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\,\mathrm dx\,\mathrm dy$

( $\partial E$ denotes the boundary of $E$ ), we want to find $M(x,y)$ and $L(x,y)$ such that

$\dfrac{\partial M}{\partial x}-\dfrac{\partial L}{\partial y}=1$

and then we would simply compute the line integral. As the hint suggests, we can pick

$\begin{cases}M(x,y)=\dfrac x2\\\\L(x,y)=-\dfrac y2\end{cases}\implies\begin{cases}\dfrac{\partial M}{\partial x}=\dfrac12\\\\\dfrac{\partial L}{\partial y}=-\dfrac12\end{cases}\implies\dfrac{\partial M}{\partial x}-\dfrac{\partial L}{\partial y}=1$

The line integral is then

$\displaystyle\frac12\int_{\partial E}-y\,\mathrm dx+x\,\mathrm dy$

We parameterize the boundary by

$\begin{cases}x(t)=5\cos t\\y(t)=17\sin t\end{cases}$

with $0\le t\le2\pi$ . Then the integral is

$\displaystyle\frac12\int_0^{2\pi}(-17\sin t(-5\sin t)+5\cos t(17\cos t))\,\mathrm dt$

$=\displaystyle\frac{85}2\int_0^{2\pi}\sin^2t+\cos^2t\,\mathrm dt=\frac{85}2\int_0^{2\pi}\mathrm dt=85\pi$

###

Notice that $x^{2/3}+y^{2/3}=4^{2/3}$ kind of resembles the equation for a circle with radius 4, $x^2+y^2=4^2$ . We can change coordinates to what you might call "pseudo-polar":

$\begin{cases}x(t)=4\cos^3t\\y(t)=4\sin^3t\end{cases}$

which gives

$x(t)^{2/3}+y(t)^{2/3}=(4\cos^3t)^{2/3}+(4\sin^3t)^{2/3}=4^{2/3}(\cos^2t+\sin^2t)=4^{2/3}$

as needed. Then with $0\le t\le2\pi$ , we compute the area via Green's theorem using the same setup as before:

$\displaystyle\iint_E\mathrm dx\,\mathrm dy=\frac12\int_0^{2\pi}(-4\sin^3t(12\cos^2t(-\sin t))+4\cos^3t(12\sin^2t\cos t))\,\mathrm dt$

$=\displaystyle24\int_0^{2\pi}(\sin^4t\cos^2t+\cos^4t\sin^2t)\,\mathrm dt$

$=\displaystyle24\int_0^{2\pi}\sin^2t\cos^2t\,\mathrm dt$

$=\displaystyle6\int_0^{2\pi}(1-\cos2t)(1+\cos2t)\,\mathrm dt$

$=\displaystyle6\int_0^{2\pi}(1-\cos^22t)\,\mathrm dt$

$=\displaystyle3\int_0^{2\pi}(1-\cos4t)\,\mathrm dt=6\pi$

3 0

2 years ago

Which equation can be used to solve for c? Triangle A B C is shown. Angle A C B is 90 degrees and angle A B C is 35 degrees. The

aliina [53]

Answer:

The correct option is B) $c=\frac{5}{\cos35^0}$ .

Step-by-step explanation:

Consider the provided information.

Angle A C B is 90 degrees and angle A B C is 35 degrees.

The required figure is shown below:

We need to find the value of c.

Use the trigonometric function: $\cos\theta=\frac{Adjacent}{Hypotenuse}$

$\cos35^0=\frac{5}{c}$

$c=\frac{5}{\cos35^0}$

Hence, the correct option is B) $c=\frac{5}{\cos35^0}$ .

4 0

1 year ago

Read 2 more answers

Katrina has the option of an 8-year non-subsidized student loan of $32,000 at an annual interest rate of 3.5% or an 8-year subsi

Fiesta28 [93]

Answer: katrina will pay less interest on the 8-year non subsidized student loan

The total interest paid on the non-subsidized loan is USD 8,959

The total interest paid on the subsidized loan is USD 11,520

Step-by-step explanation:

Subsidized loan at 4.5% for 8years will be 4.5/100 × 32000 x 1/12= USD120/month, USD 1440/year. For 8 years, USD11,520

For non subsidized student loan at 3.5% will be 3.5/100 x 32000x 1/12= USD 93.33/month, USD 1,119per year and USD 8,959 for 8 years

6 0

1 year ago

In a (blank),one ratio compares a part to a whole

Verdich [7]

Answer:

In a part-to-whole ratio, one ratio compares a part to a whole.

3 0

2 years ago

Use the formula A=bh , where A is the area, b is the base length, and h is the height of the parallelogram, to solve this proble

Jlenok [28]

33 in because 1056 divided by 32 is 33
hope you pass your test

3 0

1 year ago

Read 2 more answers