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stiv31 [10]
3 years ago
5

What is a possible scale factor for the dilation of the quadrilateral ABCD?

Mathematics
1 answer:
slamgirl [31]3 years ago
6 0

Answer:

\dfrac {1}{2}

Step-by-step explanation:

According to the diagram

if

The center is o

Then Distance from O to B'C'=1

Distance from O to BC=1+1=2

\therefore\sf Ratio\:is\:1:2.

<h3><u>More</u><u> </u><u>to</u><u> </u><u>know</u><u>:</u><u>-</u></h3><h3><u>Formulas</u><u> </u><u>related</u><u> </u><u>to</u><u> </u><u>surface</u><u> </u><u>area</u><u> </u><u>and</u><u> </u><u>volume</u><u>:</u><u>-</u></h3>

\begin{array}{|c|c|c|}\cline{1-3}\bf Shape&\bf Volume\ formula&\bf Surface\ area formula\\\cline{1-3}\sf Cube&\tt l^3}&\tt 6l^2\\\cline{1-3}\sf Cuboid&\tt lbh&\tt 2(lb+bh+lh)\\\cline{1-3}\sf Cylinder&\tt {\pi}r^2h&\tt 2\pi{r}(r+h)\\\cline{1-3}\sf Hollow\ cylinder&\tt \pi{h}(R^2-r^2)&\tt 2\pi{rh}+2\pi{Rh}+2\pi(R^2-r^2)\\\cline{1-3}\sf Cone&\tt 1/3\ \pi{r^2}h&\tt \pi{r}(r+s)\\\cline{1-3}\sf Sphere&\tt 4/3\ \pi{r}^3&\tt 4\pi{r}^2\\\cline{1-3}\sf Hemisphere&\tt 2/3\ \pi{r^3}&\tt 3\pi{r}^2\\\cline{1-3}\end{array}

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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
A box contains $7.05 in nickels, dimes, and quarters. There are 42 coins in all, and the sum of the numbers of nickels and dimes
german
I BELIEVE that there are 22 quarters, 13 dimes, and 7 nickels I’m super super sorry if wrong
3 0
3 years ago
The center of a circle is at the origin. and endpoint of a diameter of the circle is at (-3, -4. how long is the diameter of the
Aleks [24]
The diameter is 10:

To do this we must use the distance formula:

Distance =√(x2−x1)^2+(y2−y1)^2

So, if we substitute in our values for the origin and endpoint (origin is 1 values, endpoint is 2)

D=✓(-4-0)^2+(-3-0)^2

Simplified, this is

D=✓16+9
D=✓25
D=5

so, the distance from the center of the crcle to the endpoint is 5 (making the radius)

multiply by two, and the diameter of the circle is 10 :)
4 0
3 years ago
Convert 12°34'56” to decimal degree form. Round your answer<br> to four decimal places.
Vlada [557]

Answer:

12.57555 rounded to 12.5756

Step-by-step explanation:

34'=34*1/60=0.56°

56''=56/1/3600=0.01555°

12°+0.56°+0.01555°=12.57555°

7 0
3 years ago
Read 2 more answers
I GIVEEEDDE BRAINLILSTTT
Natasha_Volkova [10]

Answer:

\frac{1}{5}

Step-by-step explanation:

To find the scale factor of a dilation, we first find two corresponding sides. For example, the top edge of the black shape corresponds with the top edge of the purple shape. The top edge of the black shape measures 15 units while the top edge of the purple shape measures 3 units

Now, we divide the length taken from the figure by the length taken from the original shape. In this case, we divide the length of 3 by the length of 15.

3/15=\frac{1}{5}

I hope this helps!

5 0
3 years ago
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