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allochka39001 [22]
2 years ago
15

(pages 2-7)

Mathematics
1 answer:
user100 [1]2 years ago
7 0

Answer:

The answer is here

Step-by-step explanation:

https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/gtp--praxis-math--lessons--geometry/a/gtp--praxis-math--article--congruence-and-similarity--lesson

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Find the exact length of the curve. 36y2 = (x2 − 4)3, 5 ≤ x ≤ 9, y ≥ 0
IrinaK [193]
We are looking for the length of a curve, also known as the arc length. Before we get to the formula for arc length, it would help if we re-wrote the equation in y = form.

We are given: 36 y^{2} =( x^{2} -4)^3
We divide by 36 and take the root of both sides to obtain: y = \sqrt{ \frac{( x^{2} -4)^3}{36} }

Note that the square root can be written as an exponent of 1/2 and so we can further simplify the above to obtain: y =  \frac{( x^{2} -4)^{3/2}}{6} }=( \frac{1}{6} )(x^{2} -4)^{3/2}}

Let's leave that for the moment and look at the formula for arc length. The formula is L= \int\limits^c_d {ds} where ds is defined differently for equations in rectangular form (which is what we have), polar form or parametric form.

Rectangular form is an equation using x and y where one variable is defined in terms of the other. We have y in terms of x. For this, we define ds as follows: ds= \sqrt{1+( \frac{dy}{dx})^2 } dx

As a note for a function x in terms of y simply switch each dx in the above to dy and vice versa.

As you can see from the formula we need to find dy/dx and square it. Let's do that now.

We can use the chain rule: bring down the 3/2, keep the parenthesis, raise it to the 3/2 - 1 and then take the derivative of what's inside (here x^2-4). More formally, we can let u=x^{2} -4 and then consider the derivative of u^{3/2}du. Either way, we obtain,

\frac{dy}{dx}=( \frac{1}{6})( x^{2} -4)^{1/2}(2x)=( \frac{x}{2})( x^{2} -4)^{1/2}

Looking at the formula for ds you see that dy/dx is squared so let's square the dy/dx we just found.
( \frac{dy}{dx}^2)=( \frac{x^2}{4})( x^{2} -4)= \frac{x^4-4 x^{2} }{4}

This means that in our case:
ds= \sqrt{1+\frac{x^4-4 x^{2} }{4}} dx
ds= \sqrt{\frac{4}{4}+\frac{x^4-4 x^{2} }{4}} dx
ds= \sqrt{\frac{x^4-4 x^{2}+4 }{4}} dx
ds= \sqrt{\frac{( x^{2} -2)^2 }{4}} dx
ds=  \frac{x^2-2}{2}dx =( \frac{1}{2} x^{2} -1)dx

Recall, the formula for arc length: L= \int\limits^c_d {ds}
Here, the limits of integration are given by 5 and 9 from the initial problem (the values of x over which we are computing the length of the curve). Putting it all together we have:

L= \int\limits^9_5 { \frac{1}{2} x^{2} -1 } \, dx = (\frac{1}{2}) ( \frac{x^3}{3}) -x evaluated from 9 to 5 (I cannot seem to get the notation here but usually it is a straight line with the 9 up top and the 5 on the bottom -- just like the integral with the 9 and 5 but a straight line instead). This means we plug 9 into the expression and from that subtract what we get when we plug 5 into the expression.

That is, [(\frac{1}{2}) ( \frac{9^3}{3}) -9]-([(\frac{1}{2}) ( \frac{5^3}{3}) -5]=( \frac{9^3}{6}-9)-( \frac{5^3}{6}-5})=\frac{290}{3}


8 0
3 years ago
What is the gif of nine and fifty two
Novosadov [1.4K]
I assume you mean 9+52,
If so you can add the 9 and the 2 together:
9+2 = 11

Then you can add the 11 on to the 50 to get the final answer:
50+11 = 61

Hope this helps! :)
4 0
3 years ago
The expression 1.7j-3.4 factored is?
marshall27 [118]
1.7(j-2)


Hope it helped :)
3 0
3 years ago
The Martinez family went to the movies and paid $60 for two adult tickets and three children’s tickets. The Wright family consis
Pepsi [2]

Answer:

a = $13.5

Step-by-step explanation:

Let a = adult tickets

Let c = children tickets

Translating the word problem into an algebraic equation;

<u>For the Martinez family;</u>

2a + 3c = $60

<u>For the Wright family;</u>

3a + 5c = $95.5

Thus, the simultaneous equations are;

2a + 3c = $60 ..........equation 1

3a + 5c = $95.5 .........equation 2

We would use substitution method to solve;

From equation 2, we make a the subject of formula;

3a = 95.5 - 5c

a = (95.5 - 5c)/3

<em>Substituting the value of "a" into equation 1, we have;</em>

2[(95.5-5c)/3] + 3c = 60

Multiplying all through by 3;

2(95.5 - 5c) + 9c = 180

191 - 10c + 9c = 180

191 - c = 180

c = 191-180

c = $11

To find the value of a;

2a +3c = 60

<em>Substituting the value of "c" into the equation, we have;</em>

2a + 3(11) = 60

2a + 33 = 60

2a = 60 - 33

2a = 27

a = \frac{27}{2}

a = $13.5

<em>Therefore, the cost of an adult movie ticket is $13.5. </em>

5 0
2 years ago
How to do long division
fomenos

Answer:

steps: divide, multiply, subtract, and bring down.

Step 1. Calculate how many times the number outside the division bar goes into the first number inside the bar. Step 2. Put the answer on top of the bar. Step 3. Multiply the number outside the division bar by the number at the top of the bar.

4 0
3 years ago
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