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Alenkasestr [34]
3 years ago
11

Write the standard form of the equation of the circle with endpoints of a diameter at the points (7,2) and (-9,5).

Mathematics
1 answer:
lys-0071 [83]3 years ago
5 0

Answer:

(x+1)^2 + (y- 3.5)^2 =64

Step-by-step explanation:

We are given that  endpoints of a diameter at the points (7,2) and (-9,5).

Center of the circle is the mid point of diameter

So, first find the mid point of diameter

Formula : x=\frac{x_1+x_2}{2} , y=\frac{y_1+y_2}{2}

(x_1,y_1)=(7,2)\\(y_1,y_2)=(-9,5)

Substitute the values in formula

x=\frac{7-9}{2} , y=\frac{2+5}{2}

x=-1 , y=3.5

So, center of circle = (h,k )=(-1,3.5)

To find length of diameter :

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

(x_1,y_1)=(7,2)\\(y_1,y_2)=(-9,5)

d=\sqrt{(-9-7)^2+(5-2)^2}

d=\sqrt{265}

Length of radius = r = \frac{Diameter}{2}=\frac{\sqrt{256}}{2}

Standard form of the equation of the circle : (x -h)^2 + (y- k)^2 = r^2

(h,k )=(-1,3.5)

r=\frac{\sqrt{256}}{2}

Equation of the circle : (x+1)^2 + (y- 3.5)^2 =(\frac{\sqrt{256}}{2})^2

Equation of the circle : (x+1)^2 + (y- 3.5)^2 =64

Hence  the standard form of the equation of the circle with endpoints of a diameter at the points (7,2) and (-9,5) is  (x+1)^2 + (y- 3.5)^2 =64

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Answer:

a) 96.64% probability that maximum speed is at most 50 km/h

b) 24.67% probability that maximum speed is at least 48 km/h

c) 86.64% probability that maximum speed differs from the mean value by at most 1.5 standard deviations

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

\mu = 46.8, \sigma = 1.75

A. What is the probability that maximum speed is at most 50 km/h?

This is the pvalue of Z when X = 50. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{50 - 46.8}{1.75}

Z = 1.83

Z = 1.83 has a pvalue of 0.9664

96.64% probability that maximum speed is at most 50 km/h.

B. What is the probability that maximum speed is at least 48 km/h?

This is 1 subtracted by the pvalue of Z when X = 48.

Z = \frac{X - \mu}{\sigma}

Z = \frac{48 - 46.8}{1.75}

Z = 0.685

Z = 0.685 has a pvalue of 0.7533

1 - 0.7533 = 0.2467

24.67% probability that maximum speed is at least 48 km/h.

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Z = -1.5 has a pvalue of 0.0668

0.9332 - 0.0668 = 0.8664

86.64% probability that maximum speed differs from the mean value by at most 1.5 standard deviations

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