The results of the two-sample t-test, there does not appear to be enough evidence to support the claim that the mean GPA of night students is larger than the mean GPA of day students.
To test the claim that the mean GPA of night students is larger than the mean GPA of day students, we can perform a two-sample t-test. The null hypothesis is that there is no difference in the mean GPAs between the two groups, while the alternative hypothesis is that the mean GPA of night students is larger than the mean GPA of day students.
Using the given data, we can calculate the test statistic as follows:
t = (2.74 - 2.73) / sqrt((0.07^2 / 31) + (0.03^2 / 18)) = 0.45
The degrees of freedom for this test are 31 + 18 - 2 = 47. Using a t-table or calculator, we can find that the p-value for this test is approximately 0.326.
Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. This means that we do not have enough evidence to support the claim that the mean GPA of night students is larger than the mean GPA of day students at the 0.05 significance level.
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6. It takes Fletcher 4 weeks to finish 2 paintings. At this rate, how long will it take him to finish 4 paintings?
week(s).
Answer: 8 weeks
Step-by-step explanation:
Put the function y=8x2+16x+20
in vertex form f(x)=a(x-h)2+k
and determine the values of a,h
and k
Answer:
a=8
h=-1
k=19
Step-by-step explanation:
y=8(x^2+2x) + 20
completing the square:
y=8(x^2+2x+1)+20-1
y=8(x+1)^2+19
a=8, h=-1, k=19
The coordinates of a triangle are (1, 4), (3,5), and (0,8). The coordinates of the translated triangle are (3,5), (5, 6), and (2, 9). Find the translation vector.
Since the coordinates of the translated triangle are (3,5), (5, 6), and (2, 9), the translation vector is (x, y) → (x + 2, y + 1).
What is a translation?In Mathematics, the translation a geometric figure or graph to the right simply means adding a digit to the value on the x-coordinate of the pre-image while the translation a geometric figure or graph upward simply means adding a digit to the value on the y-coordinate (y-axis) of the pre-image.
By translating the pre-image of this triangle vertically upward by 1 unit and horizontally right by 2 units, the coordinates of the image include the following:
(x, y) → (x + 2, y + 1)
(1, 4) → (1 + 2, 4 + 1) = (3, 5).
(3, 5) → (3 + 2, 5 + 1) = (5, 6).
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Write the following complex number in rectangular form. 14(cos 135 deg + i * sin 135 deg)
Answer:
-7√2 + 7√2 i.
Step-by-step explanation:
To convert a complex number from polar form to rectangular form, we use the following formula:
a + bi = r(cos θ + i sin θ)
where a and b are the rectangular coordinates, r is the modulus or magnitude, and θ is the argument or angle.
In this case, we have:
r = 14
θ = 135°
So, we can plug these values into the formula and simplify:
a + bi = 14(cos 135° + i sin 135°)
a + bi = 14(-√2/2 + i * (-√2/2))
a + bi = -7√2 + 7√2 i
Therefore, the complex number in rectangular form is -7√2 + 7√2 i.
The spinner below shows 5 equally sized slices. Mal spun the dial 1000 times and got the following results.
Outcome White Grey Black
Number of Spins 389 406 205
Fill in the table below. Round your answers to the nearest thousandth.
The experimental probability that the next spin falls in a black region is given as follows:
0.205 = 20.5%.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
205 out of the previous 1000 trials fell in the black region, hence the experimental probability is given as follows:
p = 205/1000
p = 0.205.
p = 20.5%.
Missing InformationThe problem asks for the experimental probability that the next spin falls in a black region.
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In triangle SCU shown below, points T and O are on SU
and CU, respectively. Segment OT is drawn so
that ZC ZOTU.
C
S
T
U
If TU=4, OU = 5, and OC = 7, what is the length
of ST?
The length of ST is 5.6 units.
What are similar triangles?Similar triangles are a set of given triangles which have some common relations when their corresponding properties are compared. Though they are similar, but not congruent.
Considering the given information, we have on comparison;
TU/ SU = OU/ CU
4/ SU = 5/ 12
5SU = 4 x 12
= 48
SU = 48/5
= 9.6
So that;
SU = ST + TU
9.6 = ST + 4
ST = 5.6
The length of ST is 5.6 units.
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find the perimeter of the figure below.
The perimeter of the figure is 8x+16
What is perimeter?Perimeter is a math concept that measures the total length around the outside of a shape. We can calculated perimeter is a shape by adding all the sides together.
The perimeter of circle is 2πr, where r is the radius, perimeter of a square is 4l and a parallelogram is 2(l+b)
Perimeter of a rectangle is expressed as;
P = 2(l+b)
P= 2(3x+6+x+2)
P= 2(4x+8)
P= 8x+16
Therefore the perimeter of the figure is 8x+16
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Sketch the graphs of the following equations y=x+2, y=-x+2 and y=|x|+2
The required graph of the given equation y=x+2, y=-x+2, and y=|x|+2 has been shown.
The first equation, y = x + 2, represents a straight line with slope 1 and y-intercept 2. The graph passes through the point (0, 2) and has a positive slope, which means it slants upwards from left to right.
The second equation, y = -x + 2, also represents a straight line, but with slope -1 and y-intercept 2. The graph passes through the point (0, 2) and has a negative slope, which means it slants downwards from left to right.
The third equation, y = |x| + 2, represents the absolute value function of x translated up 2 units. The graph of y = |x| is a V-shape with its vertex at the origin, and when we add 2 to every y-coordinate, the graph is shifted upwards by 2 units. The resulting graph has a V-shape with its vertex at (0, 2).
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What number is missing 243, 81, 27, 3, 1
Answer:
9
Step-by-step explanation:
1 · 3 = 3
3 · 3 = 9
9 · 3 = 27
27 · 3 = 81
81 · 3 = 243
Write a two-column proof for the following information. Upload your proof below.
Given: M is the midpoint of CD; MD = 3
Prove: CM = 3
Statement Reason
M is the midpoint of CD | GivenMD = 3 | GivenCM = CD - MD | Segment addition postulateCD = 2MD | Definition of midpointCD = 2(3) | Substitution, using statement 2CD = 6 | SimplificationCM = 6 - 3 | Substitution, using statements 3 and 6CM = 3 | Simplification, using statement 7We are given that M is the midpoint of CD and that MD = 3. To prove that CM = 3, we need to show that CM is equal to the difference between CD and MD, and that CD is equal to twice MD because M is the midpoint of CD.
Using the segment addition postulate, we can express CM as CD - MD. From the definition of midpoint, we know that CD is twice MD, so we can substitute 2MD for CD in the expression for CM.
Simplifying the expression for CD, we get CD = 6. Substituting CD = 6 and MD = 3 into the expression for CM, we get CM = 6 - 3 = 3. Therefore, we have proven that CM is equal to 3.
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6. [3/4 Points]
PREVIOUS ANSWERS
TANFIN12 5.3.027.
The Taylors have purchased a $150,000 house. They made an initial down payment of $10,000 and secored a mortgage with interest charged at the rate of 6%/year on the unpaid balance. Interest
computations are made at the end of each month. If the loan is to be amortized over 30 years, what monthly payment will the Taylors be required to make? (Round your answer to the nearest cent)
830.37
DETAILS
What is their equity (disregarding appreciation after 5 years? After 10 years? After 20 years? (Round your answers to the nearest cent.)
5 years
$
x
10 years
$32840
20 years $74366
Due today need help (will give brainliest)
The value of ratio is,
⇒ 1 : 9
We have to given that;
Volume of cones are,
⇒ V₁ = 27 in³
⇒ V₂ = 343 in³
Hence, The value of ratio is,
⇒ V₁ : V₂
⇒ 27 : 243
⇒ 27 / 243
⇒ 9/81
⇒ 1 / 9
Thus, The value of ratio is,
⇒ 1 : 9
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Country A has an exponential growth rate of 4.3% per year. The population is currently 4,079,000, and the land area of Country A is 19,000,000,000 square yards. Assuming this growth rate continues and is exponential, after how long will there be one person for every square yard of land?
The number of years it would it take for there to be one person for every square yard is 108,325.68 years.
How many years would it take for there to be one person for every square yard?When there is one person for every square yard, it means that the population and land area are equal in value.
Number of years = (In FV / PV) / r
FV = future populationPV = present populationr = rate of growth(In 19 billion / 4,079,000) / 0.043 = 108,325.68 years
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what is the mean 0,2,4,3,1,1,5,0,4,3,2,5,3,2
Answer:
2 1/5 or 35/14
Step-by-step explanation:
The arithmetic mean is adding all the numbers up and dividing by how many # there are.
Add:
0+2+4+3+1+1+5+0+4+3+2+5+3+2 = 35
Divide:
35/14 = 2 1/5
Answer:
2.5
Step-by-step explanation:
mean=sum of all numbers/total number
=35/14
=2.5
Jonathan has a bag that contains strawberry chews, cherry chews, and peach chews.
He performs an experiment. Jonathan randomly removes a chew from the bag,
records the result, and returns the chew to the bag. Jonathan performs the
experiment 57 times. The results are shown below:
A strawberry chew was selected 4 times.
A cherry chew was selected 50 times.
A peach chew was selected 3 times.
If the experiment is repeated 300 more times, about how many times would you
expect Jonathan to remove a cherry chew from the bag? Round your answer to the
nearest whole number.
Find X and Y
X= (Simplify your answer. Do not include the degree symbol in your answer.)
Y= (Simplify your answer. Do not include the degree symbol in your answer.)
The value of variables 'x' and 'y' will be 70 degrees and 75 degrees, respectively.
The value of 'x' is calculated as,
(140 - x) / 2 = 35
Simplify the equation, then the value of 'x' is calculated as,
(140° - x) / 2 = 35°
140° - x = 70°
x = 70°
The measure of the third arc is calculated as,
⇒ 360° - 70° - 140°
⇒ 150°
The value of y is calculated as,
y + [(180° - 150°) / 2] = 90°
y + 15° = 90°
y = 90° - 15°
y = 75°
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determine the exact values of x for x^2 - 8x - 5 = 0
The exact values of x for x² - 8x - 5 = 0 are x = 4 + √21 or x = 4 - √21
To solve for x in the equation x² - 8x - 5 = 0, we can use the quadratic formula, which is:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
Using this formula, we can identify the values of a, b, and c in the equation x² - 8x - 5 = 0.
Here, a = 1, b = -8, and c = -5.
Substituting these values into the formula, we get:
x = (-(-8) ± √((-8)² - 4(1)(-5))) / 2(1)
Simplifying this expression, we get:
x = (8 ± √(64 + 20)) / 2
x = (8 ± √84) / 2
Now, we can simplify the square root of 84 as 2√21:
x = (8 ± 2√21) / 2
We can further simplify this expression by factoring out 2 from the numerator:
x = 2(4 ± √21) / 2
Canceling out the 2s, we get:
x = 4 ± √21
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(11.04 LC) 96 x16 X X 113 6 6 / 6 / 7 16 O 10 inches 16 inches Length of Hyacinth Bulbs inches inches X17 XXX- Length (in inches) What is the sum of the lengths of the longest and shortest bulbs? 7 / 7 / 8 X XX-6 X 9 9/9/3/10
The sum of the lengths of the longest and shortest bulb = 16 1/3 inches
The correct answer is an option (c)
Here, the line plot represents the Length of Hyacinth Bulbs inches.
We need to find the sum of the lengths of the longest and shortest bulb.
We knnow that a Line plot displays data as check marks above a number line that represents the frequency of each value.
From the attached line plot, we can observe that the length of the longest bulbs = 10 in. and the length of the shortest bulbs = 6 1/3
So, the sum of the lengths of the longest and shortest bulb would be,
10 + 6 1/3 = 16 1/3
Therefore, the correct answer is an option (c)
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find the volume of the shaded solid below
The volume of the shaded portion of the cone is V = 621.667 inches³
Given data ,
Let the volume of the shaded region be V
Now , the volume of the larger cone be A
Let the volume of the smaller cone be B
Volume of Cone = ( 1/3 ) πr²h
V = A - B
And , A = ( 1/3 ) ( 9 )² ( 25 )
On simplifying , we get
A = ( 1/3 ) ( 81 ) ( 25 )
A = 675 inches³
And , B = ( 1/3 ) ( 4 )² ( 10 )
B = ( 160/3 ) inches³
So , Volume of shaded region V = A - B
V = 675 inches³ - ( 160/3 ) inches³
V = ( 1,865/3 ) inches³
V = 621.667 inches³
Hence , the volume of cone is 621.667 inches³
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Tan 5pie / 6
It’s for 7.6.7 so if you have the other answers thatd be great
The value of tan( 5π/6) is -1/√3
What is trigonometric function?The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
The values of function of angles depends on the quadrant they fall on.
First quadrant ( 0-90°) = all functions are positive
Second quadrant ( 91°-180°) = only sine is positive
Third quadrant ( 181°- 270°) = only tan is positive
Fourth quadrant ( 271° -360°) = only cos is positive
Pi is a measurement of angle in radians and it is equivalent to 180°
therefore;
5π/6 = 5×180/6
= 150°
the equivalent of 150° is
180° -150 = 30°
since 150 falls on the second quadrant, tan of 150 will be negative.
Tan 150 = -tan 30
= -1/√3
therefore the value of tan 150 = -1/√3
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HELPPP THIS IS DUE ASAP NO ROCKY!!
The length of BE in given figure of ABCD rectangle is 7.2 units.
Given that the length of the rectangle ABCD is = 12 units
Breadth of the rectangle ABCD is = 9 units
Area of the rectangle ABCD = Length*Breadth = 12*9 = 108 square units.
In another way, In triangle ABC, BE is the height on the side AC which is diagonal of the rectangle.
So diagonal AC = square root of (12² + 9²) = square root of 225 = 15 units.
So the area of triangle ABC = (1/2)*BE*AC = (1/2)*BE*15 = 15BE/2 square units.
Since diagonal of a rectangle divides the rectangle into two similar triangles.
So the area of ABCD = 2*area of triangle ABC = 2*(15BE/2) = 15BE square units.
According to condition now,
15BE = 108
BE = 108/15 = 7.2 units
Hence, the length of BE is 7.2 units.
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4. Find u, u and o² for the random variable X that has the probability density
for
0
elsewhere
G(x)=6x(1-x)
0
-
The values of μ, μ² and σ² of the random variable X in the probability density function are μ = -1/2, μ² = -3/10 and σ² = -11/20
Calculating μ, μ² and σ² of the random variable X that in the probability density functionGiven that
g(x) = 6x(1 - x)
The expected value, μ is calculated as
[tex]\mu = \int\limits^a_b {x \cdot f(x)} \, dx[/tex]
So, we have
[tex]\mu = \int\limits^1_0 {x \cdot 6x(x - 1)} \, dx[/tex]
Integrate
μ = 1.5x⁴ - 2x³
So, we have
μ = 1.5(1)⁴ - 2(1)³ + 1.5(0)⁴ - 2(0)³
Evaluate
E(x) = μ = -1/2
The variable, μ² is calculated as
[tex]\mu^2 = \int\limits^a_b {x^2 \cdot f(x)} \, dx[/tex]
So, we have
[tex]\mu^2 = \int\limits^1_0 {x^2 \cdot 6x(x - 1)} \, dx[/tex]
Integrate
μ² = 3x⁴(4x - 5)/10
So, we have
μ² = 3(1)⁴(4(1) - 5)/10 - 3(0)⁴(4(0) - 5)/10
Evaluate
E(x²) = μ² = -3/10
Calculating the variance σ², we have
σ² = E(x²) - E(x)²
This gives
σ² = (-3/10) - (-1/2)²
Evaluate
σ² = -11/20
Hence, the variance is -11/20
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Identify the hidden line.
A.
B.
C.
D.
E.
The identity of each of the hidden lines are:
A is a continuous thin single arrow head line.
B is a continuous thick short dashes line.
C is a long thin dashed and single dashed line.
D is a continuous thin double arrow head line.
E is a continuous thin zig-zag line.
Types of line.A line is either a straight or curved path that joins to points of reference. Or it extends from one point to another point. There are various types of lines especially used in technical drawing, and each has its major purpose of use.
From the types of line given in the question, the identity of the hidden lines are;
i. A is a continuous thin single arrow head line.
ii. B is a continuous thick short dashes line.
iii. C is a long thin dashed and single dashed line.
iv. D is a continuous thin double arrow head line.
v. E is a continuous thin zig-zag line.
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Justin wants to buy a frame for his painting. The frame is 22 1/2
inches wide and 26 ½ inches long around the exterior. The
wood all around the frame is exactly 2 ½ inches wide. How
big does Justin's painting need to be to fit in this frame?
Justin's painting will be 20 inches wide and 24 inches long to ensure it fits in the frame width of 2 1/2 inches.
How to find the dimensions of the frameIn order to discern the ideal dimensions for Justin's painting, let us start by subtracting the wood frame width (2 1/2 inches) from the outside width and length of the frame set at 22 1/2 in and 26 1/2, respectively.
total interior width = 22 1/2 - 2 1/2 = 20 in
interior length = 26 1/2 - 2 1/2 = 24 in
This leaves a total interior width of 20 in and an interior length of 24 in; ensuring Justin's painting is sized properly to fit within the specified parameters of the prearranged frame.
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5. Find Y.
Trapezoid Worksheet
Identify Which kind of trapezoid
Answer:
y = 28
Step-by-step explanation:
(20 + 5y + 4)/2 = 3y - 2
5y + 24 = 6y - 4
y = 28
24. Calculate the median income from the following income distribution. Also find out the lowest income of highest 10% earners. Income (Rs. 000) (Less than) No. of earners LO 5 5 10 25 15 60 20 100 25 125 40 156
The median income from the given income distribution would be Rs. 34, 000.
The lowest income of the highest 10 % would be Rs. 40, 000.
How to find the median income ?Find the total number of earners sampled :
= 5 + 25 + 60 + 100 + 125 + 140 + 150 + 156
= 761 people
This means that the median income would be at :
= ( 761 + 1 ) / 2
= 381 position
The income at the 381 st position is Rs. 34, 000 so this must be the median.
The lowest of the highest 10 % would be the 90th percentile which would have the position of:
= 90 % x 761
= 685 position
The income at the 685 th position is Rs. 40, 000.
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Discussion Topic How and what you spend your money on affects the tax deductions and credits that you're eligible for. Visit the IRS website to see descriptions of the various tax credits and tax deductions currently offered. If you could propose a tax credit or a deduction for an expense that isn't currently on the list, what would it be? Would you propose it as a credit or a deduction? Why you think individuals should receive a credit or deduction for this expense?
An attractive tax credit or deduction that can be considered is a reward for those who expend in renewable energy sources, such as solar power systems, wind turbines, or geothermal technology.
The credit or deduction for this expense?This payment can take the form of a tax credit, awarding taxpayers with a dollar-for-dollar payoff to lessen their tax debt for the year they initiated the investment.
These investments in green energy assets facilitate diminishing greenhouse gases emissions, providing ecological welfares by dropping one's carbon footprint. Moreover, this spending can lower an individual’s obligation on traditional energy options, economizing them long term on utility costs. In turn, supplying these incentives encourages people to invest in recyclable energy resources, ultimately leading to a decrease in global climate change elements alongside propelling the globe into a more sustainable future.
Overall, any brand new regulation on taxation allowance needs to bear mindful inspection by assembly members, carefully examining financial and revenue implications.
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Fill in the missing data(round p value to 4 decimal places, mean values to 1 decimal place and other answers to 2 decimal places
Time in Examination Rooms (minutes)
Physician 1 Physician 2 Physician 3 Physician 4
32
28
58872882
36
26
30
27
32
23
Treatment
Physician 1
Physician 2
Physician 3
Physician 4
Total
Source
638538
Treatment
Error
Total
29
31
43
33
Mean
SS
FESSESS8
30.4
33.5
28.5
32.7
27.6
17
31
30
28
31
28
Click here for the Excel Data File
Fill in the missing data. (Round your p-value to 4 decimal places, mean values to 1 decimal place, and other answers to 2 decimal
places.)
28
30
#88888BD
35
33
25
33
35
42
Standard
Deviation
One-Factor ANOVA
df
MS
3.85
6.55
4.07
5.63
F
p-value
The mean and standard deviation of each physician's examination room times are:
Physician 1: Mean = 28.71, n= 7, Standard Deviation = 4.67Physician 2: Mean = 33.5, n =6, Standard Deviation = 3.37Physician 3: Mean = 29.13, n= 8, Standard Deviation = 2.82Physician 4: Mean = 33, n = 7, Standard Deviation = 4.28What is the mean and standard deviation of each data?The mean and standard deviation of each physician's examination room times are found using the formulas below:
Mean = sum of values/number of values
Standard deviation, σ = √(Σ(x-μ)² / N)
where
σ is the standard deviation,Σ is the sum of the squared differences between each data point and the mean,x is each data point,μ is the mean,n is the number of values.For Physician 1:
Sum: 36 + 26 + 27 + 30 + 27 + 32 + 23 = 201
n: 7
Mean = 201 / 7
Mean = 28.71
σ = √{(36-28.71)² + (26-28.71)² + (27-28.71)² + (30-28.71)² + (27-28.71)² + (32-28.71)² + (23-28.71)²} /7
σ = 4.67
For Physician 2:
Sum: 32 + 33 + 29 + 31 + 43 + 33 = 201
n: 6
Mean = 201 / 6
Mean = 33.5
σ = √{(32-33.5)² + (33-33.5)² + (29-33.5)² + (31-33.5)² + (43-33.5)² + (33-33.5)²} / 6
σ = 3.37
For Physician 3:
Sum: 17 + 31 + 30 + 28 + 31 + 38 + 28 + 30 = 233
n = 8
Mean = 233 / 8
Mean = 29.13
σ = √{(17-29.13)² + (31-29.13)² + (30-29.13)² + (28-29.13)² + (31-29.13)² + (38-29.13)² + (28-29.13)² + (30-29.13)²} / 8
σ = 2.82
For Physician 4:
Sum: 28 + 35 + 33 + 25 + 33 + 35 + 42 = 231
n = 7
Mean = 231 / 7
Mean = 33
σ = √{(28-33)² + (35-33)² + (33-33)² + (25-33)² + (33-33)² + (35-33)² + (42-33)²} / 7
σ = 4.28
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I need the answer because I already got it wrong
The time required for the visit to the doctor should be more than 27 minutes.
It is given that the dose given by doctor is 14 mg radioactive dye. Dye remaining after 16 minutes is 4.25 mg.
Assuming the radioactive dye to decay exponentially then the amount present in body can be represented by the following expression
[tex]A(t) = A_0e^-^k^t[/tex] where, A(t) is the amount present after time t, k is a constant & t is the time elapsed
Initially,[tex]4.25=14e^-^1^6^k[/tex]
k=0.074/min
To calculate time such that 2mg of radioactive dye remains in the body substituting the values
[tex]2=14e^-^t^0^.^0^7^4[/tex]
t = 26.29 minutes.
Hence, the time required for doctor's visit must be more than 27 minutes to avoid the alarm.
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Use the information provided to write the standard form equation.
Answer:
1. [tex](x-3)^2+(y+15)^2=13[/tex]
2. [tex](x+9)^2+(y+14)^2=5[/tex]
3. [tex](x+3)^2+(y+11)^2=\sqrt{51}[/tex]
4. [tex](x+3)^2+(y-0)^2=25[/tex]
For Questions 1-2:Step 1: Formula
The standard form of a circle is [tex](x-h)^2+(y-k)^2=r^2[/tex].
The h represents the x value of the coordinates of the circle's center and the k is the y value of the coordinates. The r is the radius.
So in the first problem h is going to be 3 and k is going to be -15. (In this problem the signs are flipped, so if you have a positive number it will be negative in the formula)
After inserting the values in the formula you have your answer of [tex](x-3)^2+(y+15)^2=r^2[/tex]. Now let us find the radius
Step 2: Find the radius
To find the radius of the first two problems what you can do is you can use the distance formula.
To use the distance formula you find have to identify the (x1, y1) and the (x2, y2)
In the first problem, the (x1, y1) would be the center of the circle. The reason for this is that on a graph you have to read it left to right (kinda like English) so to find the values you have to imagine what values would come first, in this case, the center point would come first.
So let us input the values in the formula. The formula is [tex]d = \sqrt{(x_{2}-x_{1})^{2} +(y_{2}-y_{1})^{2} }[/tex]. This is the distance formula, this formula will be used in this problem.
Okay so the x1 is 3 and the y1 is -15. The x2 is 0 and the y2 is -13. So lets us input these values.
[tex]d = \sqrt{(0-3)^{2}+(-13--15)^2 } \\\\ = \sqrt{(-3)^{2}+(2)^2} \\\\=\sqrt{9+4}\\\\=\sqrt{13}\\\\= 3.605[/tex]
3.605 is our radius.
Now let us write our finished equation.
[tex](x-3)^2+(y+15)^2=\sqrt{13} ^2[/tex] or [tex](x-3)^2+(y+15)^2=13[/tex].
Doing these steps for the next problem gives us our second answer of [tex](x+9)^2+(y+14)^2=5[/tex]
For Questions 3-4:Step 1: Midpoint Formula
To solve this problem you need to use the midpoint formula. The midpoint formula is [tex]\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2}[/tex]. In the first problem, the x1 would be -10, the y1 would be -13. The x2, on the other hand, is 4 and the y2 is -9. Now find the center of the circle.
Step 2: Find the center
To find the center of the circle we need to insert the values of x and the values of y into the midpoint formula. We can do this because they gave us the diameter and the diameter goes through the middle of the circle. So we can do this.
[tex]\frac{-10 + 4}{2} ,\frac{-13+-9}{2}\\\\\frac{-6}{2} ,\frac{-22}{2}\\\\(-3,-11)[/tex]
So (-3, -11) is the center of the circle.
Step 3: Find the radius
Now that we have the center of the circle we can use the distance formula to find the radius. We already have two of the ends of the radius so we can simply use the distance formula with the center and the ends.
[tex]d = \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }\\\\=\sqrt{(4--3)^2+(-9--11)^2 }\\\\= \sqrt{(7)^2+(2)^2 }\\\\=\sqrt{49+4 }\\\\=\sqrt{51}[/tex]
Step 4: Write your final equation
Know we can insert all the values we have into the standard form of a circle equation.
[tex](x-h)^2+(y-k)^2=r^2\\\\(x+3)^2+(y+11)^2=\sqrt{51} ^2[/tex] or [tex](x+3)^2+(y+11)^2=\sqrt{51}[/tex]. This is our answer.
Doing these steps for the next problem gives us our answer of [tex](x+3)^2+(y-0)^2=25[/tex]
