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3 years ago

50 POINTS EACH

Mathematics

1 answer:

Andrews [41]3 years ago

3 0

Since s and t are complimentary, that means ∠s + ∠t = 90

From the picture, we also know that ∠t + ∠LGM = 90

Set them equal to each other:

∠s + ∠t = ∠t + ∠LGM, solve:

∠s = ∠LGM

Now using the fact that tangent = opposite/adjacent:

tan (∠s) = $\frac{y}{h}$

tan (∠s) = $\frac{h}{x}$

Set them equal to each other to get:

$\frac{y}{h}$ = $\frac{h}{x}$

Solve:

$h^{2}= xy\\h = \sqrt{xy}$

Part B:

Using tan (∠s) = $\frac{y}{h}$

We get: $h = \frac{3}{tan38} = 3.84$ meters

Hope that helps!

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Answer:

the answer is C

Step-by-step explanation:

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Y=2/5+3; u gotta find two points, count the rise over run and find the y-intercept

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3 years ago

A tank contains 180 gallons of water and 15 oz of salt. water containing a salt concentration of 17(1+15sint) oz/gal flows into

Stels [109]

Let A(t) denote the amount of salt (in ounces, oz) in the tank at time t (in minutes, min).

Salt flows in at a rate of

$\dfrac{dA}{dt}_{\rm in} = \left(17 (1 + 15 \sin(t)) \dfrac{\rm oz}{\rm gal}\right) \left(8\dfrac{\rm gal}{\rm min}\right) = 136 (1 + 15 \sin(t)) \dfrac{\rm oz}{\min}$

and flows out at a rate of

$\dfrac{dA}{dt}_{\rm out} = \left(\dfrac{A(t) \, \mathrm{oz}}{180 \,\mathrm{gal} + \left(8\frac{\rm gal}{\rm min} - 8\frac{\rm gal}{\rm min}\right) (t \, \mathrm{min})}\right) \left(8 \dfrac{\rm gal}{\rm min}\right) = \dfrac{A(t)}{180} \dfrac{\rm oz}{\rm min}$

so that the net rate of change in the amount of salt in the tank is given by the linear differential equation

$\dfrac{dA}{dt} = \dfrac{dA}{dt}_{\rm in} - \dfrac{dA}{dt}_{\rm out} \iff \dfrac{dA}{dt} + \dfrac{A(t)}{180} = 136 (1 + 15 \sin(t))$

Multiply both sides by the integrating factor, $e^{t/180}$ , and rewrite the left side as the derivative of a product.

$e^{t/180} \dfrac{dA}{dt} + e^{t/180} \dfrac{A(t)}{180} = 136 e^{t/180} (1 + 15 \sin(t))$

$\dfrac d{dt}\left[e^{t/180} A(t)\right] = 136 e^{t/180} (1 + 15 \sin(t))$

Integrate both sides with respect to t (integrate the right side by parts):

$\displaystyle \int \frac d{dt}\left[e^{t/180} A(t)\right] \, dt = 136 \int e^{t/180} (1 + 15 \sin(t)) \, dt$

$\displaystyle e^{t/180} A(t) = \left(24,480 - \frac{66,096,000}{32,401} \cos(t) + \frac{367,200}{32,401} \sin(t)\right) e^{t/180} + C$

Solve for A(t) :

$\displaystyle A(t) = 24,480 - \frac{66,096,000}{32,401} \cos(t) + \frac{367,200}{32,401} \sin(t) + C e^{-t/180}$

The tank starts with A(0) = 15 oz of salt; use this to solve for the constant C.

$\displaystyle 15 = 24,480 - \frac{66,096,000}{32,401} + C \implies C = -\dfrac{726,594,465}{32,401}$

So,

$\displaystyle A(t) = 24,480 - \frac{66,096,000}{32,401} \cos(t) + \frac{367,200}{32,401} \sin(t) - \frac{726,594,465}{32,401} e^{-t/180}$

Recall the angle-sum identity for cosine:

$R \cos(x-\theta) = R \cos(\theta) \cos(x) + R \sin(\theta) \sin(x)$

so that we can condense the trigonometric terms in A(t). Solve for R and θ :

$R \cos(\theta) = -\dfrac{66,096,000}{32,401}$

$R \sin(\theta) = \dfrac{367,200}{32,401}$

Recall the Pythagorean identity and definition of tangent,

$\cos^2(x) + \sin^2(x) = 1$

$\tan(x) = \dfrac{\sin(x)}{\cos(x)}$

Then

$R^2 \cos^2(\theta) + R^2 \sin^2(\theta) = R^2 = \dfrac{134,835,840,000}{32,401} \implies R = \dfrac{367,200}{\sqrt{32,401}}$

and

$\dfrac{R \sin(\theta)}{R \cos(\theta)} = \tan(\theta) = -\dfrac{367,200}{66,096,000} = -\dfrac1{180} \\\\ \implies \theta = -\tan^{-1}\left(\dfrac1{180}\right) = -\cot^{-1}(180)$

so we can rewrite A(t) as

$\displaystyle A(t) = 24,480 + \frac{367,200}{\sqrt{32,401}} \cos\left(t + \cot^{-1}(180)\right) - \frac{726,594,465}{32,401} e^{-t/180}$

As t goes to infinity, the exponential term will converge to zero. Meanwhile the cosine term will oscillate between -1 and 1, so that A(t) will oscillate about the constant level of 24,480 oz between the extreme values of

$24,480 - \dfrac{267,200}{\sqrt{32,401}} \approx 22,995.6 \,\mathrm{oz}$

and

$24,480 + \dfrac{267,200}{\sqrt{32,401}} \approx 25,964.4 \,\mathrm{oz}$

which is to say, with amplitude

$2 \times \dfrac{267,200}{\sqrt{32,401}} \approx \mathbf{2,968.84 \,oz}$

6 0

2 years ago

Jacob the dog eats dry food that contains 324kcal/cup. Jacob eats 3.5 cups a day. The food contains 3.6 g of fat in 100 kcals of

PtichkaEL [24]

Answer:

His daily fat intake is 40.824g.

His daily Vitamin D intake is 290.304 IU.

Step-by-step explanation:

The first step to solve this problem is finding the daily kcal intake of the dogs.

Each cup has 324 kcal, and he eats 3.5cups a day. So:

1 cup - 324 kcals

3.5 cups - x kcals

$x = 324*3.5$

$x = 1134$ kcals.

His daily intake is of 1134 kcals.

The food contains 3.6 g of fat in 100 kcals of energy. what is his daily fat intake?

There are 3.6g of fat in 100 kcals of energy. How many g of fat are there in 1134 kcals?

3.6g - 100 kcals

xg - 1134 kcal

$100x = 1134*3.6$

$x = \frac{1134*3.6}{100}$

$x = 11.34*3.6$

$x = 40.824$ g

His daily fat intake is 40.824g.

The food contains 25.6 IU of Vitamin D3 in 100 kcals of energy. what is his daily Vitamin D inake?

Similar logic as above.

25.6IU - 100 kcals

x IU - 1134 kcal

$100x = 1134*25.6$

$x = \frac{1134*25.6}{100}$

$x = 11.34*25.6$

$x = 290.304$ IU

His daily Vitamin D intake is 290.304 IU.

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3 years ago

Geometry:

Kamila [148]

A statement that can be expressed in if-then form is a conditional statement.
A sentence beginning with <em>if </em>poses a condition that has to be met.

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3 years ago