Part A: Use the Pythagorean Theorem to derive the standard equation of the circle, with center at (a, b) and a point on the circle at (x, y). Show all necessary math work.
Part B: If (a, b) = (5, –2) and c = 10, determine the domain and range of the circle.
Part C: Is the point (10, 2) inside the border of the circle if (a, b) = (5, –2) and c = 10? Explain using mathematical evidence.
Answers
The distance to the point is less than c=10, so the point is inside the circle.
Given that, (a, b) = (5, –2) and c = 10.
What is domain and range of the function?The domain and range are defined for a relation and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs respectively. For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:
Domain = the set of all x-coordinates = {1, 2, 3, 4}
Range = the set of all y-coordinates = {2, 3}
Part A:
Use of the Pythagorean theorem gets you to the equation for a circle in essentially one step:
Sum of squares of sides = square of hypotenuse
(x -a)²+(y -b)² = c² circle centered on (a, b) with radius c.
Part B:
The circle will be defined for values of x in the domain f ± c, and for values of y in the range b ± c.
domain: 5 ± 10 = [5, -5]
Range: -2 ±10 = [8, -11]
Part C:
The distance from point (10, 2) to (a, b)=(5, -2) is
c² = (10-5)² +(-2-2)²
c² = 25+16
c² = 41
c=√41
c=6.4
The distance to the point is less than c=10, so the point is inside the circle.
Therefore, the distance to the point is less than c=10, so the point is inside the circle.
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Related Questions
For the experiment of rolling an ordinary pair of dice, find the probability that the sum will be less than 6 or greater than 9.
The probability that the sum will be less than 6 or greater than 9 is
Answers
The probability that the sum is 5/9.The additive dice roll has 36 different results, as explained. A sum of 9 is rolled using the numbers 6 (and vice versa) and 3 or 5 and 4. (and vice versa).
What is the likelihood of rolling two dice together?The additive dice roll has 36 different results, as explained. A sum of 9 is rolled using the numbers 6 (and vice versa) and 3 or 5 and 4. (and vice versa). Given that this is feasible 4 times out of 36, the probability calculated by adding the two dice is 4/36, or 1/9.
I am assuming that you are rolling two dice.
Probability of even: 18 "evens" out of 36 total outcomes = 18/36
→(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5),(4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2), (6,4), (6,6)
Probability of greater than 9: 6 ">9s" out of 36 total outcomes = 6/36
→(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)
Probability of both evens and >9s: 4 "both" out of 36 total outcomes = 4/36
→(4,6), (5,5), (6,4), (6,6)
evens OR >9s NOT both
18/36 + 6/8 - 4/36 = 20/36 = 5/9.
The probability that the sum is 5/9.
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A small rectangle has a width of 5cm and a length of 7cm. using the same ratio, what will be the length of a bigger rectangle it its width is 90cm?
Answers
We know that for a width of 5 cm we have a length of 7 cm,
Length Width
5 cm ————> 7cm
90 cm ————> x
We multiply in X this data:
5x = 90 * 7
x = 630/5
x = 126 cm
So with all of this we found out that the length of the rectangle with a width of 90 cm is 126 cm.
I need help with number 2...
Answers
For isosceles triangle RST the measure of angle T is 34 degree
In this question, we have been given a isosceles triangle RST with RS = RT.
So, by isosceles triangle theorem,
angle S = angle T
So, angle T = (3x - 2)°
We know that the sum of all angles of triangle is 180°
∠R + ∠S + ∠T = 180°
9x + 4 + 3x - 2 + 3x - 2 = 180°
15x + 4 - 4 = 180°
15x = 180°
x = 12
So, 3x - 2 = 3(12) - 2
= 34°
Therefore, for triangle RST the measure of angle T is 34 degree
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determine the degree, leading coefficient and constant term or each polynomial
1. x+5
degree:______
leading coefficient:_____
constant term:____
Answers
For the given polynomial the degree is 1, the leading coefficient is 1, and the constant term is 5.
What are polynomials?A polynomial in mathematics is an expression made up of coefficients and indeterminates and involves only the operations of multiplication, addition, subtraction, and non-negative integer exponentiation of variables.
The given polynomial is x+5.
(1) The degree is the highest power of the polynomial.
Degree = 1
(2) The leading coefficient is 1.
(3) The constant term is 5
Therefore, the given polynomial degree is 1, the leading coefficient is 1, and the constant term is 5.
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Factor: 64x³ + 125y³
Answers
64x³+125y³
= (4x+5y)( 16x²-20xy+25y²)
a³ + b³ = ( a + b )( a² - ab + b² )
64x³ + 125y³
= ( 4x )³ + ( 5y )³
= ( 4x+5y )[ (4x)²- (4x)(5y)+(5y)²]
= (4x+5y)( 16x²-20xy+25y² )
Find the dimensions of the rectangle with the largest area; it has two vertices on the x-axis and two vertices above the x-axis on the graph of
y = 5 − x^2.
Answers
Answer:
(2/3)√15 wide by 10/3 tall, approx 2.582 by 3.333
Step-by-step explanation:
You want the dimensions of the largest rectangle that fits under the graph of y = 5 -x² above the x-axis.
DimensionsThe graph of y=5-x² is symmetrical about the y-axis, so the width of it will be x -(-x) = 2x and the height will be 5-x².
AreaThe area will be the product of the width and height:
A = WH
A = (2x)(5 -x²) = 10x -2x³
Maximum areaThe area will be a maximum where the derivative of the area function is zero.
dA/dx = 10 -6x² = 0
x² = 10/6
x = (√15)/3
The corresponding rectangle dimensions are ...
width = 2x = (2/3)√15
height = 5 -x² = 5 -5/3 = 10/3
The rectangle with largest area is (2/3)√15 wide by 10/3 tall.
6 books weigh 1.260 kg. How many books will weigh 3.150 kg?
Answers
Answer: 15 books
Step-by-step explanation:
Given that the weight of 6 books is 1.260 kg
⟹ weight of each book is [tex]\frac{1.260}{6}[/tex] = 0.210kg
Let the number of books that will weigh 3.150 kg be x
as we know that
weight of x books is 0.210x kg
⟹0.210x=3.150
⟹x= [tex]\frac{3.150}{0.210}[/tex] = 15
The number of books is 15
15 books will weigh 3.150 kg.
Given the weight of 6 books is 1.260 kg
⟹ weight of each book is
1.260/6
=0.210 kg
Let x be the total number of books that weigh 3.150 kg.
as we are aware
weight of x books is 0.210x kg
⟹0.210x=3.150
⟹x= 3.150/0.210
=15
So, the number of books is 15.
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A scientist mixes water (containing no salt) with a solution that contains 20% salt. she wants to obtain 120 ounces of the mixture that is 15 % salt. How many ounces of water and how many ounces of the 20% salt solution should she use?
Answers
Answer:Say w= the amount of water in oz, then the 60% of the solution is 250 - w
Set up the equation:
Use distributive property first
0.60(250-w) = 0.15 x 250
Isolate to solve for w
150- 0.60w= 37.5
-0.60w= 37.5-150
0.60w= -112.5
w= 187.5 ounces
250-187.5= 62.5 ounces salt solution
Therefore, she must use 187.5 ounces of water and 62.5 ounces of salt solution for her experiment.
Hope this helps!
Step-by-step explanation:
A caterer charges a $40 fee plus an additional $6.25 per person. She determines the total cost, C, for x people by using the equation shown above. How much will a party cost for 80 people?
Answers
$40+$6.25(80)=c
Use I=PRT
p=12,000
r=4%
t=3year
find I
Answers
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$12000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &3 \end{cases} \\\\\\ I = (12000)(0.04)(3) \implies I = 1440[/tex]
HELP PLEASE QUICKLY!!!!
Answers
The expression that shows the result of applying the distributive property is 21y - 7/11 (option D).
What is the expression?The distributive property states that an expression that has this form a(b + c) can be solved as a x (b + c) which becomes ab + ac.
Thus, to solve an expression using the distributive property, multiply the term in the brackets by the term outside the brackets. Multiplication is the process of determining the product of two or more numbers. The sign that represents multiplication is x.
7(3y - 1/11)
= 7 x (3y - 1/11)
= 21y - 7/11
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(-2,6), m = 4 !!!!!!!!!!!!
Answers
Answer:
y=4x+14
Step-by-step explanation:
Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 220 numerical entries from the file and r = 52 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1.
(i) Test the claim that p is less than 0.301. Use = 0.05.
A button hyperlink to the SALT program that reads: Use SALT.
(a) What is the level of significance?
State the null and alternate hypotheses.
H0: p < 0.301; H1: p = 0.301
H0: p = 0.301; H1: p ≠ 0.301
H0: p = 0.301; H1: p > 0.301
H0: p = 0.301; H1: p < 0.301
Correct: Your answer is correct.
(b) What sampling distribution will you use?
The Student's t, since np > 5 and nq > 5.
The standard normal, since np > 5 and nq > 5.
The Student's t, since np < 5 and nq < 5.
The standard normal, since np < 5 and nq < 5.
Correct: Your answer is correct.
What is the value of the sample test statistic? (Round your answer to two decimal places.)
(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)
Answers
The hypothesis are given as follows:
H0: p = 0.301; H1: p ≠ 0.301.
a) The level of significance is of 0.05.
b) The sampling distribution is given by: The standard normal, since np > 5 and nq > 5.
The test statistic is of: z = -2.09.
c) The p-value is of: 0.0366.
What are the hypothesis tested?At the null hypothesis, it is tested if there is not enough evidence to conclude that the proportion has changed, hence:
[tex]H_0: p = 0.301[/tex]
At the alternative hypothesis, it is tested if there is enough evidence to conclude that the proportion has changed, hence:
[tex]H_1: p \neq 0.301[/tex]
What is the test statistic?The test statistic is given by the equation presented as follows:
[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
The parameters of the equation are listed as follows:
[tex]\overline{p}[/tex] is the sample proportion.p is the proportion tested at the null hypothesis.n is the sample size.The values of these parameters for this test are:
[tex]\overline{p} = \frac{52}{220} = 0.2364, p = 0.301, n = 220[/tex]
Hence the value of the test statistic is of:
[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
[tex]z = \frac{0.2364 - 0.301}{\sqrt{\frac{0.301(0.699)}{220}}}[/tex]
z = -2.09.
What is the p-value?We have a two-tailed test, as we are testing if the proportion is different of a value.
Hence, using a z-distribution calculator, with z = -2.09, the p-value is of 0.0366.
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What is the equation of the line that passes through the point (2,-7) and has a slope of -4
Answers
Answer: y+7=-4(x-2)
Step-by-step explanation:Point slope form is: y - y1 = m(x-x1)2 = x1-7 = y1 substituteAnswer: y+7=-4(x-2)
2. Determine if the diagram is a function.
Why or why not?
Please
Answers
Answer:
IT IS A FUNCTION
It is specifically a Surjective/ onto Mapping/function
Step-by-step explanation:
surjective or onto function is when y has an image in x
Answer:
The graph is a function
Step-by-step explanation:
-For every input (x) there is only one output value (y)
Since the function does not have two of the same x values for several inputs, its considered a function. If the x values do not repeat values, or if you graph the function and it does not cross on more than one point, it is also considered a function.
Ex.
X: 2, 4, 7 , 2, 5 This is not a function. More than one input for each output.
Y: 1, 2, 4, -3, 7
Ex.
X: 5, 6, -5, 8 this is a function. Exactly one input for every output.
Y: 1, 4, 5, 9
A number line going from negative 20 to positive 20.
What is the absolute value of –20?
Use the number line to answer the question.
–20
0
20
40
Answers
Answer:
20
Step-by-step explanation:
Absolute value is the distance from zero. Absolute value is always positive.
| -20 | is positive. It is 20 units from 0. So it is 20.
| -20 | = 20
(Be careful, absolute value doesn't just flip the sign to the opposite.
| 20 | = 20 also because the distance from 0 to 20 is also 20.)
Find the slope of the line
Answers
Answer:
-3
Step-by-step explanation:
we sure y2-y1/x2-x1 to find slope
6+6/-4+0
12/-4
-3 is the slope
Hopes this helps please mark brainliest
5x-3(x-3)=-5+4x+6 x=?????????
Answers
x = 4
Explanation:
5x - 3x + 9 = -5 + 4x + 6
2x + 9 = 1 + 4x
2x = 8
x = 4
I would appreciate it if you mark my answer as brainliest.
Answer:
[tex]\frac{7}{4}[/tex] or 1 [tex]\frac{3}{4}[/tex]
Step-by-step explanation:
I am not sure if you meant 6x or 6+x at the end of the equation. I went with what was written.
5x -3(x-3) = -5 +4x +6x Distribute the -3
5x -3x + 9 = -5 + 4x + 6x Combine like terms
2x + 9 = -5 + 10x Subtract 2x from both sides
9 = -5 + 8x Add 5 to both sides
14 = 8x Divide both sides by 5
[tex]\frac{14}{8}[/tex] = x
[tex]\frac{7}{4}[/tex] o r 1 [tex]\frac{3}{4}[/tex]
The measurements of two sides of a triangle are 12cm and 17cm Which side length could be the measurement of the third side of the triangle
Answers
Using the triangle rule, the length of the third side falls between 5 cm and 29 cm.
In the given question,
The measurements of two sides of a triangle are 12 cm and 17 cm.
We have to find which side length could be the measurement of the third side of the triangle.
We know that,
The sum of the length of any two sides is always greater than the third side.
The given sides of triangles are 12 cm and 17 cm.
Hence, the length of the third side should be shorter than the total of the other two sides,
12 + 17 = 29 cm.
Considering that the third side must be less than the difference between the first two sides,
17 – 12 = 5 cm
So, the length of the third side falls between 5 cm and 29 cm.
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Created a painting with an area of 48 in.² and a length of 6 inches they create a second painting with an area of 64 in.² it has the same width as the first painting what is the length of the second painting
Answers
The length of the second rectangle is 8 in
How to find the length of the second painting?Since we have painting with an area of 48 in.² and a length of 6 inches, since it is a rectangle, its area is A = LW where
L = length of rectangle = 6 in and W = width of rectangleSo, making Wsubject of the formula, we have
W = A/L
Since
A = 48 in² and L = 6 inSubstituting these into the equation, we have
W = A/L
W = 48 in²/6 in
W = 8 in
Now, since the second painting with an area of 64 in.² it has the same width as the first painting, since it is also a rectangle, its area is A = LW where
L = length of rectangle and W = width of rectangleSo, making L subject of the formula, we have
L = A/W
Since
A = 64 in² and W = 8 inSubstituting these into the equation, we have
L = A/W
L = 64 in²/8 in
L = 8 in
So, the length of the second rectangle is 8 in
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need help with this asap first answer will get brainliest
Answers
Answer:
x° = 34°
Step-by-step explanation:
First, find the interior angle supplementary to the given exterior angle 124°.
I will label it y.
124° + y° = 180°
y° = 180° - 124°
y° = 56°
Next, solve for x° by equating the sum of the interior angles of the triangle to 180° and plugging in the y-value that we just solved for.
x° + y° + 90° = 180°
x° + y° = 90°
x° + 56° = 90°
x° = 90° - 56°
x° = 34°
Answer:
x = 34---------------------------
Exterior angle in the triangle is the sum of remote interior angles124 = x + 90x = 124 - 90x = 34Given that capital is fixed, what do you figure is the marginal cost or additional cost of producing another:
haircut
sandwich
Ferris Wheel ride
Answers
The marginal cost is the labor utilized in order to provide an additional cost of a haircut.
What is the marginal cost?The marginal cost is the extra cost of generating one additional unit of output.
The example of a haircut, the marginal cost is the labor utilized in order to provide an additional haircut; employees offer their time, which is valued in pay, resulting in the marginal cost of labor.
Thus, in all circumstances, such as the sandwich and Ferris Wheel, there will be an additional labor cost because labor is necessary to make each of the items.
Change in cost / Change in quantity Equals Marginal cost.
The firm will incur an increased labor expense because of one more haircut, and the laborer will be paid more because of the rise in one more quantity of haircut delivered. This pay rise is the marginal cost.
Thus, the marginal cost is the labor utilized in order to provide an additional cost of a haircut.
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Two buses leave a station at the same time and travel in opposite directions. On bus travels 17 km/h faster than the other. If the two buses are 836 kilometers apart after 4 hours, what is the rate of each bus
Answers
Step-by-step explanation:
speed = distance / time
to get distance, we have to multiply by time.
the speed the 2 busses are moving seat from each other is
x + (x + 17) = 2x + 17
x = the speed of one bus
x + 17 = the speed of the other bus.
so, after 4 hours the distance between them is
(2x + 17) × 4 = 836
2x + 17 = 209
2x = 192
x = 96 km/h
bus 1 travels at 96 km/h.
bus 2 travels at 96+17 = 113 km/h.
India's hourly wage is $10.50. If she
regularly works 40 hours per week, what is
her regular weekly pay?
Answers
Her regular weekly pay is $420
Step-by-step explanation:
10 hours per week: $10.50 x 10 = $105.00
20 hours per week: $105.00 x 2 = $210.00
40 hours per week: $210.00 x 2 = $420.00
Una grúa está elevando 5 /7 de los 224 kg que puede elevar como máximo ¿Cuántos kilos está elevando?
Answers
The number of kilograms that the crane is lifting, based on the maximum it can lift and the proportion being lifted is 160 kilos
How to find the number of kilos?The number of kilos that the crane is lifting can be found by the formula :
= Maximum kilogram capacity x Proportion of maximum being lifted
Maximum kilogram capacity = 224 kg
Proportion of maximum being lifted = 5 / 7
The kilos being lifted by the crane is therefore :
= 224 x 5/7
= 1, 120 / 7
= 160 kilos
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8.7% increase of 525 what is answer
Answers
Answer:
Step-by-step explanation:
570.675
Fewer than %97 of adults have a cell phone. In a reputable poll of 1160 adults, %88
said that they have a cell phone. Find the value of the test statistic
Answers
The value of the test statistic of the hypotheses is; -17.969
How to find the test statistic?
The test statistic is defined as a number, calculated from a statistical test, which is used to find if your data could have occurred under the null hypothesis.
We are given;
Sample size; n = 1160
Sample proportion; p^ = 88% = 0.88
Population proportion; p = 97% = 0.97
The claim is about the population proportion (p) in the hypotheses. Since we are told that fewer than 97% of adults have a cell phone, then the claim is; p < 0.97
The null hypothesis is expressed as;
H₀: p = 0.97
The alternative hypothesis is expressed as;
Hₐ: p < 0.97
The value of the test-statistic is gotten from the formula;
z = (p^ - p)/(√(p(1 - p)/n))
z = (0.88 - 0.97)/(√(0.97(1 - 0.97)/1160))
z = -17.969
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Please help me verify. I am stuck.
Answers
The trigonometric identity presented in this problem is proved trough the answer, reaching an answer 1 = 1.
Proving the identityThe identity is presented as follows:
[tex]\csc{\beta} + 1 = \frac{\cot{\beta}}{\sec{\beta} - \tan{\beta}}[/tex]
Then we take the right side of the denominator, and apply the subtraction of perfect squares to the denominator, as follows:
[tex]\frac{\cot{\beta}}{\sec{\beta} - \tan{\beta}} \times \frac{\sec{\beta} + \tan{\beta}}{\sec{\beta} + \tan{\beta}} = \frac{\cot{\beta}(\sec{\beta} + \tan{\beta})}{\sec^{2}{\beta} - \tan^2{\beta}}[/tex]
The following identity relates the square of the secant and of the tangent:
[tex]\tan^2{\beta} + 1 = \sec^2{\beta}[/tex]
Hence:
[tex]\sec^2{\beta} - \tan^2{\beta} = 1[/tex]
Thus the remaining expression is:
[tex]\frac{\cot{\beta}(\sec{\beta} + \tan{\beta})}{\sec^{2}{\beta} - \tan^2{\beta}} = \cot{\beta}(\sec{\beta} + \tan{\beta})[/tex]
The equations of the cotangent, secant and tangent are given as follows:
Cotangent: cosine/sine.Secant: 1/cosine.Tangent: sine/cosine.Hence:
[tex]\cot{\beta}(\sec{\beta} + \tan{\beta}) = \frac{\cos{\beta}}{\sin{\beta}}\left(\frac{1}{\cos{\beta}} + \frac{\sin{\beta}}{\cos{\beta}}\right) = \frac{1}{\sin{\beta}} + 1[/tex]
One divided by the sine is the cosecant, as follows:
[tex]\frac{1}{\sin{\beta}} = \csc{\beta}[/tex]
Hence the simplified expression is:
[tex]\csc{\beta} + 1[/tex]
Which is equivalent to the right side of the expression, hence an identity of the format 1 = 1 is reached and the identity is proven.
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HELP 50 POINT MATH QUESTION MIDDLE SCHOOL
Answers
In the proportional relationship table, x would represent distance and y, weight.
The complete table is given in the explanation below.
What is a Proportional Relationship?If two variables that have quantities that vary proportionately are x and y, then the constant of proportionality that lies between both varies can be calculated as k = y/x. The equation for the proportional relationship would be expressed as y = kx.
x represents distance (cm)
y represents weight (newtons).
Using the pair of value that is given in the table, (20, 28), the constant of proportionality, k, would be:
k = y/x = 28/20
k = 1.4
The equation would be y = 1.4x.
Use the equation to complete the table by substituting the value of each missing variables:
Row 2, when x = 55:
y = 1.4(55)
y = 77
Row 3, when y = 140:
140 = 1.4x
140/1.4 = x
100 = x
x = 100
Row 4, when x = 1:
y = 1.4(1)
y = 1.4
The table would be:
Distance (cm) Weight (newtons)
Row 1 20 28
Row 2 55 77
Row 3 100 140
Row 4 1 1.4
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Describe the graph of g?
Answers
The given points gives us a line with constant slope of -2.
What are coordinates?
A pair of numbers that use the separations between the two reference axes to define the location of a point on a coordinate plane. usually represented by the x- and y-values, respectively, (x, y).
The position of a point or a shape in a given space is determined by coordinates, which are numbers (a map or a graph ).
The points are in the form of ( x, y) , where y= g(x)
Plot these points on the graph and join the points through a line.
Refer to the attachment for the graph.
The slope is constant.
consider any 2 points:
(x,y) = (0,7)
(m,n) =(2,3)
Slope = change in y values/ change in x values
= (7-3)/(0-2) = 4/-2 = -2
The given points gives us a line with constant slope of -2.
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