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Studentka2010 [4]
3 years ago
15

There are 420 books on a bookcase with 7 shelves. Each shelf has the same number of books. How many books are on 4 shelves? 28 6

0 120 240 ...
Mathematics
1 answer:
Ghella [55]3 years ago
3 0
If we have 420 books for every 7 shelves and are looking for how many books are on 4 shelves, we need to set up fractional proportions.

We have 7 shelves for every 420 books, so we will have our first fraction like this:
7/420.
We also have 4 shelves for every x books, so our second fraction will be 4/x.

Set these fractions up as an equation.
7/420 = 4/x

To solve for x, we need to cross multiply the numerators of one fraction with the denominators of the other fraction.
In other words, multiply the top number of one fraction by the bottom number of the other question.

It should look like this:
7(x)
420(4)

7(x) = 7x
420(4) = 1680

Set your products up as an equation.
7x = 1680

Now to solve for x, we need to simplify.
Currently, x is being multiplied by 7.
To get x alone, divide 7x by 7.
Remember, what you do to one side of an equation, you must do to the other, so divide 1680 by 7.

7x / 7 = x
1680 / 7 = 240

You are left with:
x = 240.

For every 4 shelves you have 240 books.

I hope this helps!

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Answer:

a) The equation is (y - 1)² = -8 (x - 4)

b) The equation is (x - 1)²/25 + (y - 4)²/16 = 1

c) The equation of the ellipse is (x - 3)²/16 + y²/4 = 1

Step-by-step explanation:

a) Lets revise the standard form of the equation of the parabola with a

   horizontal axis

# (y - k)² = 4p (x - h), (h , k) are the coordinates of its vertex and p ≠ 0

- The focus of it is (h + p , k)

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∵ The focus is (2 , 1)

∵ focus is (h + p , k)

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∵ k = 1

∴ Its equation is (y - 1)² = 4p (x - h)

- The parabola contains point (2 , 5), substitute the coordinates of the

 point in the equation of the parabola

∴ (5 - 1)² = 4p (2 - h)

∴ (4)² = 4p (2 - h)

∴ 16 = 4p (2 - h) ⇒ divide both sides by 4

∴ 4 = p (2 - h) ⇒ (2)

- Use equation (1) to substitute h in equation (2)

∴ 4 = p (2 - [2 - p]) ⇒ open the inside bracket

∴ 4 = p (2 - 2 + p) ⇒ simplify

∴ 4 = p (p)

∴ 4 = p² ⇒ take √ for both sides

∴ p = ± 2, we will chose p = -2 because the parabola opens left

- Substitute the value of p in (1) to find h

∵ h = 2 - p

∵ p = -2

∴ h = 2 - (-2) = 2 + 2 = 4

∴ The equation of the parabola in standard form is

  (y - 1)² = 4(-2) (x - 4)

∴ The equation is (y - 1)² = -8 (x - 4)

b) Lets revise the equation of the ellipse

- The standard form of the equation of an ellipse with  center (h , k)

 and major axis parallel to x-axis is (x - h)²/a² + (y - k)²/b² = 1  

- The coordinates of the vertices are (h ± a , k )  

- The coordinates of the foci are (h ± c , k), where c² = a² - b²  

* Now lets solve the problem

∵ Its vertices are (-4 , 4) and (6 , 4)

∵ The coordinates of the vertices are (h + a , k ) and (h - a , k)  

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∴ h + a = 6 ⇒ (1)

∴ h - a = -4 ⇒ (2)

- Add (1) and (2) to find h

∴ 2h = 2 ⇒ divide both sides by 2

∴ h = 1

- Substitute the value of h in (1) or (2) to find a

∴ 1 + a = 6 ⇒subtract 1 from both sides

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∵ The foci at (-2 , 4) and (4 , 4)

∵ The coordinates of the foci are (h + c , k) , (h - c , k)

∴ h + c = 4

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∴ 1 + c = 4 ⇒ subtract 1 from both sides

∴ c = 3

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∴ 3² = 5² - b²

∴ 9 = 25 - b² ⇒ subtract 25 from both sides

∴ -16 = -b² ⇒ multiply both sides by -1

∴ 16 = b²

∵ a² = 25

∵ The equation of the ellipse is (x - h)²/a² + (y - k)²/b² = 1

∴ The equation is (x - 1)²/25 + (y - 4)²/16 = 1

c) How to identify the type of the conic  

- Rewrite the equation in the general form,  

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- Identify the values of A and C from the general form.  

- If A and C are nonzero, have the same sign, and are not equal  

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- If A and C are equal and nonzero and have the same sign, then

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- If A and C are nonzero and have opposite signs, and are not equal  

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- If either A or C is zero, then the graph is a parabola  

* Now lets solve the problem

∵ x² + 4y² - 6x - 7 = 0

∵ The general form of the conic equation is

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∴ A = 1 and C = 4

∵ If A and C are nonzero, have the same sign, and are not equal  to

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∵ x² + 4y² - 6x - 7 = 0 ⇒ re-arrange the terms

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- Lets make x² - 6x completing square

∵ 6x ÷ 2 = 3x

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∴ (x² - 6x + 9) - 9 + 4y² - 7 = 0 ⇒ simplify

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∴ (x - 3)² + 4y² = 16 ⇒ divide all terms by 16

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∴ (x - 3)²/16 + y²/4 = 1

∴ The equation of the ellipse is (x - 3)²/16 + y²/4 = 1

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