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3 years ago

How much does 2.5 L of oil cost if 2,000 mL costs $5.00? Show work

Mathematics

1 answer:

kkurt [141]3 years ago

5 0

Answer:

$6.25

Step-by-step explanation:

2000ml = 2 litres

2.5L - 2L = 500ml

So now we know that without the 500ml, it is 5 dollars.

Find out how much 500 ml cost.

2000ml = 5.00

1ml = 5.00 ÷ 2000 = 0.0025

500ml = 0.0025 x 500 = 1.25

2L + 500 ml = 2.5L (which 2.5L costs 6.25)

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there are 112 male and 78 females working at the hospital. what percentage of the hospital workforce is female

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78/190=41.05% are female

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3 years ago

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Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant below the line y=5 and betw

vfiekz [6]

First, complete the square in the equation for the second circle to determine its center and radius:

x ² - 10x + y ² = 0

x ² - 10x + 25 + y ² = 25

(x - 5)² + y ² = 5²

So the second circle is centered at (5, 0) with radius 5, while the first circle is centered at the origin with radius √100 = 10.

Now convert each equation into polar coordinates, using

x = r cos(θ)

y = r sin(θ)

Then

x ² + y ² = 100 → r ² = 100 → r = 10

x ² - 10x + y ² = 0 → r ² - 10 r cos(θ) = 0 → r = 10 cos(θ)

y = 5 → r sin(θ) = 5 → r = 5 csc(θ)

See the attached graphic for a plot of the circles and line as well as the bounded region between them. The second circle is tangent to the larger one at the point (10, 0), and is also tangent to y = 5 at the point (0, 5).

Split up the region at 3 angles θ₁, θ₂, and θ₃, which denote the angles θ at which the curves intersect. They are

θ₁ = 0 … … … by solving 10 = 10 cos(θ)

θ₂ = π/6 … … by solving 10 = 5 csc(θ)

θ₃ = 5π/6 … the second solution to 10 = 5 csc(θ)

Then the area of the region is given by a sum of integrals:

$\displaystyle \frac12\left(\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\}\left(10^2-(10\cos(\theta))^2\right)\,\mathrm d\theta+\int_{\frac\pi6}^{\frac{5\pi}6}\left((5\csc(\theta))^2-(10\cos(\theta))^2\right)\,\mathrm d\theta\right)$

$=\displaystyle 50\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\} \sin^2(\theta)\,\mathrm d\theta+\frac12\int_{\frac\pi6}^{\frac{5\pi}6}\left(25\csc^2(\theta) - 100\cos^2(\theta)\right)\,\mathrm d\theta$

To compute the integrals, use the following identities:

sin²(θ) = (1 - cos(2θ)) / 2

cos²(θ) = (1 + cos(2θ)) / 2

and recall that

d(cot(θ))/dθ = -csc²(θ)

You should end up with an area of

$=\displaystyle25\left(\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\}(1-\cos(2\theta))\,\mathrm d\theta-\int_{\frac\pi6}^{\frac{5\pi}6}(1+\cos(2\theta))\,\mathrm d\theta\right)+\frac{25}2\int_{\frac\pi6}^{\frac{5\pi}6}\csc^2(\theta)\,\mathrm d\theta$

$=\boxed{25\sqrt3+\dfrac{125\pi}3}$

We can verify this geometrically:

• the area of the larger circle is 100π

• the area of the smaller circle is 25π

• the area of the circular segment, i.e. the part of the larger circle that is bounded below by the line y = 5, has area 100π/3 - 25√3

Hence the area of the region of interest is

100π - 25π - (100π/3 - 25√3) = 125π/3 + 25√3

as expected.

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3 years ago

Esteban is reading a book with 16 pages. He has read 9 pages so far. How many pages does Esteban have left to read? Write two eq

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Answer:

7 pages

Step-by-step explanation:

Given the total number of pages in the book are 16

The number of pages esteban already read are 9

Let the number of pages left out to read be x

Total number of pages = number of pages read + number of pages to be read

16=9+x

x=16-9

x=7 pages

Therefore There are 7 pages left out to be read.

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3 years ago

Write the point-slope equation oft line that goes through (1, -1) with slope of -5.

Sever21 [200]

Answer:

y+1=-5(x-1)

Step-by-step explanation:

We need to find the slope intercept form first. So far we have this: y=-5x+b. We could plug in the x value and y value to get this : -1=-5(1)+b. b=4. Now we have the slope intecept form: y=-5x+4, and we could turn it into point slope form.

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2 years ago

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Answer:

(-2, -31)

Extra Letters.

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