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3 years ago

Can someone explain this to me please

Mathematics

1 answer:

IrinaVladis [17]3 years ago

3 0

Answer:

c. 36·x

Step-by-step explanation:

Part A

The details of the circle are;

The area of the circle, A = 12·π cm²

The diameter of the circle, d = $\overline {AB}$

Given that $\overline {AB}$ is the diameter of the circle, we have;

The length of the arc AB = Half the the length of the circumference of the circle

Therefore, we have;

A = 12·π = π·d²/4 = π· $\overline {AB}$ ²/4

Therefore;

12 = $\overline {AB}$ ²/4

4 × 12 = $\overline {AB}$ ²

$\overline {AB}$ ² = 48

$\overline {AB}$ = √48 = 4·√3

$\overline {AB}$ = 4·√3

The circumference of the circle, C = π·d = π· $\overline {AB}$

Arc AB = Half the the length of the circumference of the circle = C/2

Arc AB = C/2 = π· $\overline {AB}$ /2

$\overline {AB}$ = 4·√3

∴ C/2 = π·4·√3/2 = 2·√3·π

The length of arc AB = 2·√3·π cm

Part B

The given parameters are;

The length of $\overline {OF}$ = The length of $\overline {FB}$

Angle D = angle B

The radius of the circle = 6·x

The measure of arc EF = 60°

The required information = The perimeter of triangle DOB

We have;

Given that the base angles of the triangles DOB are equal, we have that ΔDOB is an isosceles triangle, therefore;

The length of $\overline {OD}$ = The length of $\overline {OB}$

The length of $\overline {OB}$ = $\overline {OF}$ + $\overline {FB}$ = $\overline {OF}$ + $\overline {OF}$ = 2 × $\overline {OF}$

∴ The length of $\overline {OD}$ = 2 × $\overline {OF}$ = The length of $\overline {OB}$

Given that arc EF = 60°, and the point 'O' is the center of the circle, we have;

∠EOF = The measure of arc EF = 60° = ∠DOB

Therefore, in ΔDOB, we have;

∠D + ∠B = 180° - ∠DOB = 180° - 60° = 120°

∵ ∠D = ∠B, we have;

∠D + ∠B = ∠D + ∠D = 2 × ∠D = 120°

∠D = ∠B = 120°/2 = 60°

All three interior angles of ΔDOB = 60°

∴ ΔDOB is an equilateral triangle and all sides of ΔDOB are equal

Therefore;

The length of $\overline {OD}$ = The length of $\overline {OB}$ = The length of $\overline {DB}$ = 2 × $\overline {OF}$

The perimeter of ΔDOB = The length of $\overline {OD}$ + The length of $\overline {OB}$ + The length of $\overline {DB}$ = 2 × $\overline {OF}$ + 2 × $\overline {OF}$ + 2 × $\overline {OF}$ = 6 × $\overline {OF}$

∴ The perimeter of ΔDOB = 6 × $\overline {OF}$

The radius of the circle = $\overline {OF}$ = 6·x

∴ The perimeter of ΔDOB = 6 × 6·x = 36·x

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To find the equation for the line of regression where the x-axis representing the number of years after 1980 and the y-axis representing the percent increase of donations made on behalf of a non-profit organization.
$\begin{center}
\begin{tabular}{ c| c| c| c| }
 x & y & x^2 & xy \\ [1ex] 
 4 & 7.8 & 16 & 31.2 \\ 
 9 & 16.3 & 81 & 146.7 \\ 
13 & 26.2 & 169 & 340.6 \\ 
17 & 38.9 & 289 & 661.3 \\ 
21 & 49.2 & 441 & 1,033.2 \\ 
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\Sigma x=87 & \Sigma y=200.5 & \Sigma x^2=1,525 & \Sigma xy=3,641.3 
\end{tabular}
\end{center}$

Recall that the equation of the regression line is given by
$y=a+bx$
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$a= \frac{(\Sigma y)(\Sigma x^2)-(\Sigma x)(\Sigma xy)}{n(\Sigma x^2)-(\Sigma x)^2} = \frac{200.5(1,525)-87(3,641.3)}{6(1,525)-(87)^2} \\ \\ = \frac{305,762.5-316793.1}{9,150-7,569} = \frac{-11,030.6}{1,581} =-6.977$
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Thus, the equation of the regresson line is given by
$y=-6.977+2.7858x$

The graph of the regression line is attached.

Using the equation, we can predict the percent donated in the year 2015. Recall that 2015 is 35 years after 1980. Thus x = 35.

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