Answer:
See proof below.
Step-by-step explanation:
If we assume the following linear model:
And if we have n sets of paired observations 
the model can be written like this:
And using the least squares procedure gives to us the following least squares estimates 
for 
and 
for 
:
Where:
Then 
is a random variable and the estimated value is 
. We can express this estimator like this:
Where 
and if we see careful we notice that 
and 
So then when we find the expected value we got:
And as we can see is an unbiased estimator for
In order to find the variance for the estimator we have this:
And we can assume that 
since the observations are assumed independent, then we have this:
And if we simplify we got:
And with this we complete the proof required.
Answer: (20) x = 8 , (21) x = 4√3 , (22) x = 9 , (23) x = 2√3 , (24) x = 140° , (25) x = 82°
<u>Step-by-step explanation:</u>
20) Given two intersecting chords inside a circle, the product of their segments is equal.
5 * x = 4 * 10
5x = 40
x = 8
21) Given a secant and a tangent line, the product of the outside segment and the entire segment length equals the square of the tangent segment.
4 * (8 + 4) = x²
4 (12) = x²
48 = x²
√48 = √x²
4√3 = x
22) Given two secant lines, the product of their outside segments and the entire segment lengths are equal.
4 * (5 + 4) = 3 * (3 + x)
4(9) = 3(3 + x)
4(3) = 3 + x
12 = 3 + x
9 = x
(23) Given two intersecting chords inside a circle, the product of their segments is equal.
6 * 2 = x * x
12 = x²
√12 = √x²
2√3 = x
24) 
(outside arc - inside arc) = vertex angle
(30 - inside arc) = 10
30 - inside arc = 20
- inside arc = -10
inside arc = 10
outside arc + x + inside arc = 180 <em>because they form a semi circle</em>
30 + x + 10 = 180
40 + x = 180
x = 140
25) opposite angles of a quadrilateral inscribed in a circle are supplementary
98 + x = 180
x = 82
A^2+b^2=c^2 (Pythagoran theorem)
-> 30^2 + length^2 = 78^2
-> length = √6084-900 = <span>√5184 = 72 ft
=> area = length x width = 72 x 30 = 2160 ft^2</span>
