Suppose the line of best fit is being found for some data points that have an r-value of 0.876. If the standard deviation of the x-coordinates is 5.559, and the standard deviation of the y-coordinates is 7.163, what is the slope of the line to three decimal places?

Equation of parabola with focus at (0,-4) and directrix is y=4 .

As we know parabola is the locus of all the points such that distance from fixed point on the parabola to fixed line directrix is same.

The parabola is opening downwards.

Let any point on parabola is (x,y).

Distance from focus(0,-4) to (x,y) = [tex]\sqrt{(x-0)^{2} +(y+4) ^{2}}=\sqrt{x^{2}+ (y+4)^{2}}[/tex]

Distance from (x,y) to directrix, y=4 is =[tex]\left | y-4 \right |[/tex]

As these distances are equal.

[tex]\sqrt{x^{2}+ (y+4)^{2}}=\left | y-4 \right |\\{x^{2}+ (y+4)^{2}=(y-4)^{2}[/tex]

→x²+y²+8 y +16 = y² - 8 y+16

→ x² = -8 y - 8 y= -16 y   [ Cancelling y² and 16 from L.H.S and R.H.S ]

So , equation of parabola is , x²= - 16 y or f(x)= -x²/16

The Equation of parabola is [tex]f(x)=-\dfrac{x^2}{16}[/tex]

Equation of parabola with focus at [tex](0,-4)[/tex] and directrix is [tex]y=4[/tex] .

Parabola is the locus of all the points such that distance from fixed point on the parabola to fixed line directrix is same.

The parabola is opening downwards.

Let any point on parabola is [tex](x,y)[/tex].

Distance from focus [tex](0,-4)[/tex] to [tex](x,y)[/tex] = [tex]\sqrt{(x-0)^2+(y+4)^2[/tex]

[tex]d=\sqrt{x^2+(y+4)^2[/tex]

Distance from [tex](x,y)[/tex] to directrix, [tex]y=4[/tex] is =[tex]\left | y-4 \right |[/tex]

As these distances are equal.

[tex]\sqrt{x^2+(y+4)^2}=\left | y-4 \right |[/tex]

[tex]d={x^2+(y+4)^2=(y-4)^2[/tex]

[tex]x^2+y^2+8y+16=y^2-8y+16[/tex]

[tex]x^2=-8y-8y[/tex]

[tex]x^2=-16y[/tex]

[tex]y=\dfrac{-x^2}{16}[/tex]

So, the Equation of parabola is [tex]f(x)=-\dfrac{x^2}{16}[/tex]

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Answer:

square

Step-by-step explanation:

i got it right in edge

After the WW2, Britain had become weaker and its powers had started to decrease. It had lost many of its soldiers and revenues in the world war. Also it needed to focus on the development of its own country now.

so, the rebels in the colonial Africa became stronger than Britishers. So Africans were able to fight back for their freedom and land. Hence, after a while colonial powers left completely from certain parts of Africa.  

Final answer:

The area of the inscribed circle within a square with an area of 900 in² is 225π in².

Explanation:

The subject here is circle geometry, which is a part of Mathematics. Given that the area of the square is 900 in², we want to find the area of the inscribed circle. The side length a of the square can be found by taking the square root of the area, so a = √{900} = 30 inches. The diameter of the circle is the same as the side of the square, which means the radius r is half of that, so r = 30 / 2 = 15 inches. The formula for the area of a circle is A = πr², and substituting in our value of r, we get:

A = π * (15 in)² = 225π in²

If the radius of the circle will be 5, then the diameter of the circle is 10 m. Then the correct option is C.

A circle can be characterized by its center's location and its radius's length.

Let the center of the considered circle be at the (h,k) coordinate.

Let the radius of the circle be 'r' units.

Then, the equation of that circle would be:

(x - h)² + (y - k)² = r²

Marisha plots her flower garden on a computer.

She determines that the circle that defines the part of the garden that gets watered by the sprinkler is (x − 8)² + (y + 10)² = 25.

Then the equation can be written as

(x − 8)² + (y + 10)² = 5²

The radius of the circle will be 5

Then the diameter of the circle is given as

d = 2r

d = 2 x 5

d = 10 m

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[tex]\text{The formula of a distance between two points:}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\text{We have the points}\ (3,\ -4)\ \text{and}\ (3,\ -17).\ \text{Substitute:}\\\\d=\sqrt{(3-3)^2+(-17-(-4))^2}=\sqrt{0^2+(-13)^2}=\sqrt{(-13)^2}=13\\\\Answer:\ \boxed{13\ units}[/tex]

Answer:

x = 26° is the answer.

Step-by-step explanation:

We have to find the value of x from the given equation.

sin 64° = cos x

since sin 64° = 0.8987

Therefore cos x = 0.8987

[tex]x = cos^{-1}(0.8987) = 26[/tex]

x = 26°

Option: A is the correct answer.

The value of cos P is:

             A)   [tex]\dfrac{9}{41}[/tex]

We know that if two angles A and B are complementary then,

[tex]A+B=90[/tex]

i.e.

[tex]B=90-A[/tex]

Here we have angle P and angle Q are complementary i.e.

[tex]P=90-Q[/tex]

Also, we are given,

[tex]\sin Q=\dfrac{9}{41}[/tex]

We are asked to find:

[tex]\cos P[/tex]

It could also be written as:

[tex]\cos P=\cos (90-Q)\\\\i.e.\\\\\cos P=\sin Q\\\\i.e.\\\\\cos P=\dfrac{9}{41}[/tex]

Final answer:

The sum of the series 4+12+36+108+...+8748 is found by observing that it is a geometric progression with a common ratio of 3. By comparing the last term to the options, and understanding the rapidly increasing nature of geometric series, the sum is slightly larger than the last term, which only matches the option 8908.

Explanation:

To find the sum of the geometric series 4+12+36+108+...+8748, we first need to identify the common ratio (r) of the series. Each term is obtained by multiplying the previous term by the constant r. In this case, r = 12 / 4 = 36 / 12 = 108 / 36 = 3. The series is therefore a geometric progression with the first term a = 4 and common ratio r = 3.

We can then use the formula for the sum of the first n terms of a geometric series, which is Sn = a(1-rn) / (1-r), where r ≠ 1. However, we need the number of terms (n). To find n, we use the formula for the nth term of a geometric sequence, which is an = a * rn-1. We can solve for n considering the last term given: 8748 = 4 * 3n-1.

However, instead of manually finding the value of n and then the sum, let us approach the problem with the available options. The sum of a geometric series increases very rapidly due to the exponential nature of each subsequent term. Comparing the last term with the provided options, we can infer that the sum has to be a little bit greater than the last term of the series, 8748.

By inspection of the options, the only number that is slightly greater than 8748 is 8908. Therefore, to the quickest approximation and without going through detailed calculations, the sum of the series is 8908.

Answer:

[tex]\angle ABC \cong \angle XYZ[/tex]

[tex]CA \cong ZX[/tex]

Step-by-step explanation:

According to the given figures:

[tex]\angle ABC \cong \angle XYZ[/tex]

Also,

[tex]CA \cong ZX[/tex]

Therefore, the answer to the first question is Angle XYZ, and the answer to the second question is ZX.

You can deduct these answers by observing the number of line that match each pair of congruent elements.

Answer: D)  [tex]y=-5(x-5)^2+1125[/tex]

Step-by-step explanation:

Let x be the number of $2 increases in price and y be the revenue.

By using the given table , lets check all the options

A) [tex]y=(x+5)^2-1125[/tex]

For x=1, y=1045

[tex]y=(1+5)^2-1125=36-1125=-1089\\\\But\ 1045\neq-1089[/tex]

B)  [tex]y=(x-5)^2+1125[/tex]

For x=1, y=1045

[tex]y=(1-5)^2+1125=14+1125=1139\\\But\ 1045\neq1139[/tex]

C)   [tex]y=-5(x+5)^2-1125[/tex]

[tex]y=-5(1+5)^2-1125=-5(36)-1125=-1305\\\\But\ 1045\neq-1305[/tex]

D) [tex]y=-5(x-5)^2+1125[/tex]

[tex]y=-5(1-5)^2+1125=-5(16)+1125=1045[/tex]

Hence, this is the quadratic equation that models the data.

Based on the table of values, an equation of the quadratic that models the data is: D. [tex]y = -5(x-5 ) ^2+ 1125[/tex]

In Mathematics and Euclidean Geometry, the vertex form of a quadratic function is represented by the following mathematical equation:

[tex]y = a(x - h)^2 + k[/tex]

Where:

In order to determine an equation for the line of best fit or quadratic regression line that models the data points contained in the table of values, we would have to use a graphing calculator (Microsoft Excel).

Based on the scatter plot (see attachment) which models the relationship between the number of $2 increase in price (in dollars), x and revenue, y, the quadratic regression is given by:

[tex]y =-5x^2 + 50x + 1000\\\\y = -5(x^2 - 10x) + 1000\\\\y = -5(x^2 - 10x + (\frac{10}{2})^2 ) + 1000+ (\frac{10}{2})^2\\\\y = -5(x^2 - 10x + 25 ) + 1000+25\\\\y = -5(x-5 ) ^2+ 1125[/tex]

The Law of Cosines is used to find the remaining parts of a triangle when the triangle is not a right triangle and when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known.

The law of cosine states if a,b,c are the three sides of any triangle then [tex] a^{2} =b^{2} +c^{2} -2bccosA [/tex]

In the above proof in statement 8 the x in statement 5 is replaced by c cos A from statement 6.

Among the given options ,option C holds true as the reason for statement 8.

Answer:

Before we could add these numbers, 3/8 and 7/10 need a common denominator. Both 8 and 10 go into 40.

8 goes into 40 five times

3/8= (3*5)/40 = 15/40

10 goes into 40 four times

7/10= (7*4)/40= 28/40

ANSWER: A) 15/40 and 28/40

Hope this helps! :)

Step-by-step explanation:

Final answer:

The probability that a DVD will have a video or audio defect is 1.9%, calculated using the principle of inclusion-exclusion. The probability that a DVD will not have any defect is 98.1%, obtained by subtracting the probability of having a defect from 100%.

Explanation:

To solve the probability that a DVD purchased by a customer will have a video or audio defect or not, we can use the principle of inclusion-exclusion for probabilities.

According to the given data:

Part a: The probability that a DVD will have a video or audio defect (at least one type of defect) is calculated by adding the probabilities of each defect and subtracting the probability of both defects occurring, as these are counted twice.

P(either type of defect) = P(video defect) + P(audio defect) - P(both defects)

P(either type of defect) = 1.5% + 0.8% - 0.4% = 1.9%

Part b: To find the probability that a DVD will not have any defect, we subtract the probability of either defect from 100%.

P(no defect) = 100% - P(either type of defect)

P(no defect) = 100% - 1.9% = 98.1%

Therefore, the probability that a DVD will have a video or audio defect is 1.9%, and the probability that a DVD will not have any defect is 98.1%.

The difference between log(x/y) and log(x)/log(y) is that the first expression can be written as log x - log y, which is not equal to the second term.

The opposite of exponentiation is the logarithm. This indicates that the exponent to which b must be raised in order to obtain a number x is the logarithm of x to the base b. For instance, because 1000 = 10³, the logarithm of 1000 in base 10 is 3, or log₁₀ = 3.

Given:

log(x/y) and log(x) / log(y),

The formula for  log(x / y)  can be written as shown below,

log(x / y)   = log x - log y

As you can see the term is not equal to log(x) / log(y), hence they are different,

The values of the function would differ as in this expression log(x/y) you would first divide x by y and then will take the log of the result,

In the expression log(x) / log(y), you will take the log first individually and then divide the answer.

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