The proof that the Quadrilateral WXYZ is a rectangle is shown below
Proving that Quadrilateral WXYZ is a rectangle.To prove that Quadrilateral WXYZ is a rectangle, we need to show that it has four right angles.
First, we can find the slopes of each pair of opposite sides to determine if they are perpendicular.
The slope of WX is:
m(WX) = (y2 - y1)/(x2 - x1) = (-2 - 1)/(2 - (-1)) = -1
The slope of YZ is:
m(YZ) = (y2 - y1)/(x2 - x1) = (6 - 3)/(4 - 7) = -1
Since the slopes of WX and YZ are equal , we know that they are opposite sides.
Now we can use the distance formula to check if the sides of the quadrilateral are congruent. The length of WX is:
√((2 - (-1))^2 + (-2 - 1)^2) = √18
The length of YZ is:
√((7 - 4)^2 + (3 - 6)^2) = √18
The length of WY is:
√((7 - (-1))^2 + (3 - 1)^2) = √68
The length of XZ is:
√((4 - 2)^2 + (6 - (-2))^2) = √68
We can see that opposite sides WX and YZ have the same length, as do opposite sides WY and XZ.
So, the quadrilateral is a rectangle
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Susan's cell phone plan includes unlimited texting. The amount she pays for texting each
month can be described by the function y = 20, where y is the number of dollars spent on
texting as a function of x, the number of texts.
What is true about the function?
It is linear because the rate of change is zero.
It is linear because the rate of change is increasing.
It is nonlinear because the rate of change is zero.
It is nonlinear because the rate of change is
increasing.
The scale factor of the larger of two similar triangular prisms is 6. The surface area of the smaller prism is 36 ft2. Identify the surface area, rounded to the nearest tenth, of the larger prism
The surface area, rounded to the nearest tenth, of the larger prism is 1296 square feet.
If the scale factor of the larger prism to the smaller prism is 6, then the corresponding ratio of their surface areas is 6² or 36:1.
Scale factor is the ratio of the lengths of corresponding sides of two similar figures. It is used to determine how much larger or smaller one figure is compared to another, and it is often expressed as a fraction or a decimal.
We know that the surface area of the smaller prism is 36 ft², so we can set up the equation
x = 36 × 36
where x is the surface area of the larger prism.
Simplifying the equation
x = 1296 square feet
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At a grocery store a four pack of yogurt cost $3.95 how much does each container of yogurt cost explain?
Answer: ~$0.99 (0.9875)
Just divide 3.95 by 4
an oil prospector will drill a succession of holes in a given area to find a productive well. the probability that he is successful on a given trial is .2. a what is the probability that the third hole drilled is the first to yield a productive well? b if the prospector can afford to drill at most ten wells, what is the probability that he will fail to find a productive well?
a) The probability that the third hole drilled is the first to yield a productive well is given by the following sequence of events: the first two holes must be unproductive, followed by a productive third hole. The probability of each of these events happening is:
- Probability of an unproductive hole: 0.8
- Probability of two unproductive holes in a row: 0.8
On a certain map, 1/2 inch represents 75 miles. If the distance between two towns on this map is 2 1/4, what is the actual distance between the two towns?
The actual distance between the two towns is 337.5 miles.
What is distance?
Distance is the measure of how far apart two objects or locations are from each other. It is usually measured in units such as meters, kilometers, miles, or feet. Distance is a scalar quantity, meaning it has only magnitude and no direction.
If 1/2 inch on the map represents 75 miles in reality, then 1 inch on the map would represent 150 miles (twice the distance of 1/2 inch).
So, to find the actual distance between the two towns, we need to multiply the distance on the map (2 1/4 inches) by the conversion factor (150 miles per inch):
2 1/4 inches * 150 miles per inch = (9/4) inches * 150 miles per inch
Simplifying:
(9/4) inches * 150 miles per inch = 337.5 miles
Therefore, the actual distance between the two towns is 337.5 miles.
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Find the measures of angle a and B. Round to the
nearest degree.
The measure of angle A and B is 29° and 61° respectively
What is trigonometric ratio?Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.
sin(tetha) = opp/hyp
tan(tetha) = opp/adj
cos(tetha) = adj/hyp
The opposite is 6 and the adjascent = 11
Therefore tan (tetha) = 11/6 = 1.833
tetha = tan^-1( 1.833)
= 61°( nearest degree)
The sum of angle in a triangle is 180°
therefore,
angle A = 180-( 61+90)
= 180-151
= 29°
therefore the measure of angle A and B is 29° and 61° respectively.
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a simple random sample of 100 8th graders at a large suburban middle school indicated that 86% of them are involved with some type of after school activity. find the 98% confidence interval that estimates the proportion of them that are involved in an after school activity. a) (0.699, 0.941) b) (0.829, 0.834) c) (0.779, 0.941) d) (0.679, 0.891) e) (0.779, 0.741) f) none of the above
For a random sample of 100 students related to aftee school activity. The 98% confidence interval that estimates the proportion of students who are involved in an after school activity is equals to (0.779, 0.941). So, option (c) is right one.
Confidence intervals represents the variation around a statistical estimate. A confidence interval is a range of interval of estimates for an unknown parameter.
[tex]CI = \hat p ± z_ \frac{ \alpha}{2}\sqrt{\frac{ ( 1 - \hat p)\hat p}{n}}[/tex]
where [tex]\hat p[/tex]-->sample proportion
n --> sample size
We have a simple random sample of 8ᵗʰ graders at a large suburban middle school. Sample size, n = 100
Sample proportion for students who involved with some type of school activity, [tex]\hat p[/tex] = 86% = 0.86
We have to calculate the 98% confidence interval that estimates the proportion of students sample.
Level of significance, α = 98%
= 0.98
Using the normal distribution table, value of z-score for 98% of confidence interval is 2.326 . Now, plug all known values in following confidence interval formula,
[tex]CI = \hat p ± z_ \frac{ \alpha}{2}\sqrt{\frac{ ( 1 - \hat p)\hat p}{n}}[/tex]
[tex]= 0.86 ± (2.326) \sqrt{\frac{ ( 1 - 0.86)0.86}{100}}[/tex]
[tex]= 0.86 ± (2.326) \sqrt{\frac{ ( 0.14)0.86}{100}}[/tex]
[tex]= 0.86 ± 0.081 [/tex]
[tex]= (0.86 - 0.081 , 0.86 + 0.081) [/tex]
= (0.779, 0.941)
Hence, required value is (0.779, 0.941).
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Three machines are used to produce nails. The table displays the total number of nails produced by the 3 machines
over different lengths of time.
Nail Production with 3 Machines
Time (minutes)
Number of Nails
(thousands)
15
16
45
48
55
593
ON
If each machine produces nails at the same rate, how many nails can 1 machine produce in 1 hour?
nails
One machine can produce 600,000 nails in one hour.
Finding the total number of nails produced by the three machines in a minute and dividing that number by three to obtain the number of nails produced by a single machine in a minute are the first two steps in the solution to this problem.
The number of nails produced by a single machine in an hour can then be calculated by multiplying that number by 60.
Now, we add up the total number of nails produced during :
(15 + 16 + 45 + 48 + 55 + 59 + 3) / 7 = 30
To find the number of nails produced by one machine in one hour, we multiply by 60: 10,000 x 60 = 600,000
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Walking tours at a park begin every 25 minutes and bus tours begin every 45 minutes. Both tours start at 8:00 a.m. when the park opens. When is the next time the tours will start at the same time?
The next time the walking and bus tours will start at the same time is 11:45 a.m.
What is the lcm?
The LCM is multiple which is useful if fractions need to be expressed in the same name, when the other number is multiple, LCM will have the larger number:
To find out when the walking and bus tours will start at the same time, we need to find the least common multiple (LCM) of 25 and 45, which is the smallest time interval that is a multiple of both 25 minutes and 45 minutes.
The prime factorization of 25 is 5 * 5, and the prime factorization of 45 is 3 * 3 * 5. To find the LCM, we take the highest power of each prime factor that appears in either factorization, so:
LCM = 3 * 3 * 5 * 5 = 225
Therefore, the walking and bus tours will start at the same time every 225 minutes, or 3 hours and 45 minutes. To find the next time they will start at the same time, we need to add 225 minutes to the starting time of 8:00 a.m.
8:00 a.m. + 3 hours and 45 minutes = 11:45 a.m.
Hence, the next time the walking and bus tours will start at the same time is 11:45 a.m.
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If anyone is reading this, rn i would be so flipping happy if u got this for me ive been waiting for so long and got nothing please answer correctly please
Answer: The answer is A.
Step-by-step explanation: Because I am smart don't underestimate me.
Answer:
C
Step-by-step explanation: (look at attachment)
3x + 4 = -2x -2
By looking at the y-intercepts, you automatically know the answer is C.
The y-intercept of the pink line is 4 because of 3x + 4.
The y-intercept of the blue line is -2, because of -2x - 2.
Number 7. I don’t understand, what’s the fraction? How do you get fraction and the + a number.
Answer:
A
Step-by-step explanation:
Equation of a line is: y = mx + b where m = slope b = y axis intercept
To find the slope between any two of the given points :
say 18, 100 and 27, 85
m = slope = (y1-y2) / (x1-x2) = (85-100) / ( 27-18) = -15/12 = -5/3
so now you have
y = - 5/3 x + b we still need to find the value of b
use any point to calculate b
say 15, 106
106 = - 5/3 (15) + b
b = ~ 131
the equation is then y = - 5/3 x + 131 closest to answer 'A'
Any help? Please. Whoever answer it first gets brainliest!
Answer:
[tex]c + 15 > 24[/tex]
[tex]c > 9[/tex]
The additional amount will be more than $9.
21st term: 3,8,13,18 What is the indicated term
The 21st term of the sequence 3, 8, 13, 18, .. is 103
To find the indicated term in the sequence, we first need to identify the pattern followed by the sequence. It appears that each term is obtained by adding 5 to the previous term. So, we can write the general formula for the nth term of the arithmetic sequence as
a(n) = a(1) + (n-1)d
where a(1) is the first term of the sequence, d is the common difference, and n is the term number.
In this case, we have:
a(1) = 3 (the first term)
d = 5 (the common difference)
To find the 21st term, we substitute n = 21 in the formula:
a(21) = a(1) + (21-1)d
a(21) = 3 + 20(5)
a(21) = 103
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find the smallest which 108 must be multiplied to get a perfect square
Answer:
The answer is 3
Step-by-step explanation:
x×108=y
x×2²×3³=y
3×108=324
Find the measure of Angle A and round the answer to the nearest tenth.
(Show work if you can, thank you).
68.5 is the measure of Angle A and round the answer to the nearest tenth.
What is Trigonometry?Trigonometry is the area of mathematics that deals with particular angles' functions and how to use those functions in calculations. There are six popular trigonometric functions for an angle. Sine (sin), cosine (cos), tangential (tan), cotangent (cot), secant (sec), & cosecant (csc) are their names, respectively.
Here we have to use the tan ratio to find the answer.
tanθ = base/perpendicular
Given: perpendicular = 19, base = 22
tan θ = 19/22
θ = tan^-1 (19/22)
tan θ = 1.2
so the angle θ is
68.5 degree
Answer: B) 68.5 degrees
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Answer:
The measure of angle A is 40.8° to the nearest tenth.
Step-by-step explanation:
The given triangle ABC is a right triangle.
We want to find the measure of angle A, and have been given the length of the sides opposite and adjacent to angle A.
The tangent ratio of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side of that angle.
Therefore, we can use the tangent trigonometric ratio to find the measure of angle A.
[tex]\boxed{\begin{minipage}{7 cm}\underline{Tangent trigonometric ratio} \\\\$\sf \tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\end{minipage}}[/tex]
Given values:
θ = angle A = xO = side BC = 19A = side AC = 22Substitute the given values into the ratio and solve for x:
[tex]\implies \tan(x)=\dfrac{19}{22}[/tex]
[tex]\implies x=\tan^{-1}\left(\dfrac{19}{22}\right)[/tex]
[tex]\implies x=40.8150838...^{\circ}[/tex]
[tex]\implies x=40.8^{\circ}\; \sf (nearest\;tenth)[/tex]
Therefore, the measure of angle A is 40.8° to the nearest tenth.
Use the image to answer the question. A coordinate plane with four quadrants shows the x- and y-axes ranging from negative 5 to 5 in increments of 1. A solid line and a dotted line intersect each other. The equation of the solid line is x minus 5 y equals 3. The equation of the dotted line is 3 x minus 2 y equals negative 4. The intersection of both lines is shown at negative 2 on the x-axis and negative 1 on the y-axis in quadrant 3.
Review the graphs of a system of two linear equations in two variables: x−5y=7 and 3x−2y=−4. Find the solution to both equations.(1 point)
The intersection point is ()
The equations given are x - 5y = 3 and 3x - 2y = -4. To find the solution to this system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
Find the solution to both equations?One way to solve this system of equations is by substitution. We can solve one equation for x or y, and then substitute that expression into the other equation to eliminate one variable. Let's solve the first equation for x:
x - 5y = 3
x = 5y + 3
Now we can substitute this expression for x into the second equation:
3x - 2y = -4
3(5y + 3) - 2y = -4
15y + 9 - 2y = -4
13y = -13
y = -1
We can now substitute this value for y back into either equation to find the value of x:
x - 5y = 3
x - 5(-1) = 3
x + 5 = 3
x = -2
Therefore, the solution to the system of equations x - 5y = 3 and 3x - 2y = -4 is (-2, -1). This is the point where the solid line and dotted line intersect, as shown in the image.
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2. Tyler leaves his house at 7:00 a.m. to go to school. He walks for 20 minutes until he reaches his school, 1 mile from his house. The function d gives the distance d(t), in miles, of Tyler from his house t minutes after 7:00 a.m. a. Explain what d(5) = 0.25 means in this context. b. On snowy days, Tyler's school has a 2 hour delayed start time (120 minutes). The function is gives Tyler's distance s(t), in miles, from home t minutes after 7:00 a.m. with a 120 minute delayed start time. If d(5) = 0.25, then what is the corresponding point on the function s? c. Write an expression for s in terms of d. A new function, n, is defined as n(t) = d(t +60) explain what this means in terms of Tyler's distance from school.
Answer: a. In this context, d(5) = 0.25 means that 5 minutes after 7:00 a.m., Tyler is 0.25 miles away from his house. This is because the function d(t) gives the distance of Tyler from his house t minutes after 7:00 a.m.
b. If d(5) = 0.25, then we know that 5 minutes after 7:00 a.m., Tyler is 0.25 miles away from his house. If there is a 120-minute delayed start time, then Tyler will walk for 20 + 120 = 140 minutes to reach his school. We want to find the corresponding point on the function s, which gives Tyler's distance from home t minutes after 7:00 a.m. with a 120-minute delayed start time. Since Tyler walks the same distance regardless of the delayed start time, we can use the same function for s as we did for d. Therefore, s(145) = 1.25, since Tyler is 1 mile away from his house after walking for 140 minutes and then an additional 5 minutes to account for the delayed start time.
c. Since Tyler walks the same distance regardless of the delayed start time, we can express s(t) in terms of d(t) by adding 120 minutes to the time t. Therefore, s(t) = d(t + 120).
d. The function n(t) = d(t + 60) gives Tyler's distance from his house t minutes after 8:00 a.m. This is because adding 60 minutes to t corresponds to adding one hour to the time, which means that Tyler leaves his house at 8:00 a.m. instead of 7:00 a.m. Therefore, n(t) gives Tyler's distance from school one hour after he leaves his house.
Step-by-step explanation:
A rock of radioactive material has 500 atoms in it. The number of atoms decreases at a rate of 11% a day. Write an exponential function that models this situation. f(x) type your answer... (1 choose your answer... choose your answer... ✓)^x
Answer:
[tex]f(x) = 500( {.89}^{x} )[/tex]
which of the following statements is true? a high correlation is insufficient to establish causation on its own. if the two variables of a scatterplot are strongly related, this condition implies causation between the two variables. only a correlation equal to 0 implies causation. a correlation of 1 or -1 implies causation.
A high correlation is insufficient to establish causation on its own.
it is believed that 5% of all people requesting travel brochures for transatlantic cruises actually take the cruise within 1 year of the request. an experienced travel agent believes this is wrong. of 100 people requesting one of these brochures, only 3 have taken the cruise within 1 year. we want to test the travel agent's theory with a hypothesis test. if you used a significance level of 0.05, what is your decision?
Based on the given information, we can set up the following hypotheses for the hypothesis test:
Null Hypothesis (H0): The actual proportion of people taking the cruise within 1 year is equal to the believed proportion of 5%.
Alternative Hypothesis (H1): The actual proportion of people taking the cruise within 1 year is not equal to the believed proportion of 5%.
Let p be the proportion of people taking the cruise within 1 year. We can use the sample proportion, denoted as p-hat, which is calculated as the ratio of the number of people who took the cruise within 1 year (3 in this case) to the total number of people who requested the brochures (100 in this case).
Given that the significance level is 0.05, we can use a z-test to compare the sample proportion with the believed proportion of 5%. The z-test statistic is calculated as:
z = (p-hat - p) / sqrt(p * (1 - p) / n)
where n is the sample size, which is 100 in this case.
Now we can calculate the z-test statistic and compare it with the critical value for a two-tailed test at a significance level of 0.05. If the calculated z-test statistic falls outside the critical value, we would reject the null hypothesis; otherwise, we would fail to reject the null hypothesis.
Since the sample proportion p-hat is 3/100 = 0.03, and the believed proportion p is 0.05, we can substitute these values into the z-test formula:
z = (0.03 - 0.05) / sqrt(0.05 * (1 - 0.05) / 100)
Calculating the above expression, we get the value of z. We can then compare this value with the critical value for a two-tailed test at a significance level of 0.05 from a standard normal distribution table or using a statistical calculator.
If the calculated z-test statistic falls outside the critical value, we would reject the null hypothesis and conclude that the actual proportion of people taking the cruise within 1 year is different from the believed proportion of 5%. If the calculated z-test statistic falls within the critical value, we would fail to reject the null hypothesis and not conclude that the actual proportion is different from the believed proportion.
Without the actual values of the calculated z-test statistic and the critical value, we cannot provide a specific decision for this hypothesis test. Please note that hypothesis testing requires careful consideration of the sample size, significance level, and other relevant factors, and should be conducted with caution and in consultation with a qualified statistician or expert in statistical analysis.
The lengths of two sides of a triangle are 5.2 inches and 3.1 inches. Which lengths, in inches, could be the length of the third side?
The length of the third side between 2.1 inches and 8.3 inches (exclusive) could be a valid length for the third side of the triangle.
Triangle Inequality Theorem:In a triangle, the length of any side must be less than the sum of the lengths of the other two sides and greater than the difference between the lengths of the other two sides.
We can apply this rule to find the possible lengths of the third side of the triangle, given that the lengths of the two sides are 5.2 inches and 3.1 inches.
Here we have
The lengths of two sides of a triangle are 5.2 inches and 3.1 inches
Let's denote the length of the third side as x. Then, we have:
3.1 + 5.2 > x > 5.2 - 3.1
8.3 > x > 2.1
Therefore, the length of the third side x must be greater than 2.1 inches and less than 8.3 inches.
We can write this as an inequality:
2.1 < x < 8.3
Therefore,
The length of the third side between 2.1 inches and 8.3 inches (exclusive) could be a valid length for the third side of the triangle.
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what is the sales tax rate on an $8.50 purchase if the sales-tax rate is 6[tex]\frac{x}{y} 1/2[/tex]%
The sales tax rate is 6%
what is the sales tax rate on an $8.50 purchase if the sales-tax rate is 6x/y1/2%?If the sales-tax rate is 6%, the tax paid on an $8.50 purchase would be:
Tax = 0.06 x $8.50
Tax = $0.51
Therefore, the total cost of the purchase including sales tax would be:
Total cost = $8.50 + $0.51
Total cost = $9.01
The sales tax rate is 6%.
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what are the first four terms if a1=2 and an= an-1 +3?
The first four terms of the sequence are a₁ = 2, a₂ = 5, a₃ = 8, a₄ = 11.
What is arithmetic progression?The difference between every two successive terms in a sequence is the same; this is known as an arithmetic progression (AP). A good example of an arithmetic progression (AP) is the series 2, 6, 10, 14,..., which follows a pattern in which each number is created by adding 4 to the previous term.
The given sequence is defined as a₁=2 and aₙ= aₙ₋₁ + 3 for n > 1.
To find the first four terms of the sequence, we can use the recursive formula repeatedly:
a₁ = 2 (given)
a₂ = a₁ + 3 = 2 + 3 = 5
a₃ = a₂ + 3 = 5 + 3 = 8
a₄ = a₃ + 3 = 8 + 3 = 11
Therefore, the first four terms of the sequence are a₁ = 2, a₂ = 5, a₃ = 8, a₄ = 11.
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We need to simplify it. How would I do this
The simplified expression is 4(x - 3y).
What is logarithmic means ?logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8.
Using the following logarithmic identities:
log a (bc) = log a (b) + log a (c)
log a (b/c) = log a (b) - log a (c)
We can simplify the expression as follows:
2㏒4 x - 6 log4 y = 2(㏒4) x - 6(㏒4) y
= 2(㏒4) x - 2(㏒4)3 y
= 2(㏒4)(x - 3y)
Now, we can simplify further by using the fact that ㏒4 = 2:
2(㏒4)(x - 3y) = 2(2)(x - 3y) = 4(x - 3y)
Therefore, the simplified expression is 4(x - 3y).
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Using the graph, determine the coordinates of the vertex of the parabola.
Answer:
Vertex = (-3, -4)
Step-by-step explanation:
The given graph is a parabola that opens upwards.
The vertex of a parabola that opens upwards is its lowest point (minimum value).
From inspection of the given graph, the lowest point is (-3, -4).
Therefore, the vertex of the parabola is (-3, -4).
Maggie spent $18. 00 Of $30. 00 In her wallet which decimal represents the fraction of the $30. 00 Maggie spent
The decimal that represents the fraction of the $30.00 Maggie spent is 0.6.
Now, let's talk about decimals. Decimals are a way of expressing parts of a whole number in a fraction of 10. For example, 0.5 is the same as 1/2. In your situation, Maggie spent $18.00 out of $30.00. To figure out what decimal represents the fraction of the $30.00 Maggie spent, we need to divide the amount she spent by the total amount she had.
So, we can write this as a fraction:
$18.00 / $30.00
To turn this fraction into a decimal, we divide the numerator (top number) by the denominator (bottom number) using long division or a calculator.
$18.00 / $30.00 = 0.6
Another way to say this is that Maggie spent 60% of the money she had in her wallet.
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does the calculated percent fat from the experimental data in question 2 agree with the percent fat calculated from the label in question 3? why or why not? group of answer choices yes, they are the same. the two values differ by less than 1%. no, they are different. the two values differ by more than 1%.
If the experimental value is 10.1% and the label value is 10%, then the relative difference is 1%, which is equal to 1%. Therefore, you can say that they agree.
To determine if the calculated per cent fat from the experimental data in question 2 agrees with the per cent fat calculated from the label in question 3, follow these steps:
1. Calculate the per cent fat from the experimental data in question 2.
2. Calculate the per cent fat from the label in question 3.
3. Compare the two values.
If the two values are the same or differ by less than 1%, then the answer is yes, they agree. If the two values differ by more than 1%, then the answer is no, they do not agree.
Without the specific data from questions 2 and 3, I cannot provide a definite answer. However, you can use the steps provided above to determine if the calculated per cent fat from the experimental data agrees with the per cent fat calculated from the label.
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If f(x) = 5x - 6, which of these is the inverse of f(x)?
A. f^-¹(x) = x/5 +6
B. f^-¹(x) = x/5 -6
C. f^-¹(x) = x+6/5
D. F^-¹(x) = x-6/5
To find the inverse of a function, we need to swap the positions of x and y and then solve for y. In other words, we replace f(x) with y and then solve for x.
So, let's start by swapping x and y in the function f(x) = 5x - 6:x = 5y - 6
Next, we'll solve this equation for y:
x + 6 = 5y
y = (x + 6)/5
Therefore, the inverse of f(x) is f^-1(x) = (x + 6)/5, which is option C.how many 6 card hands are there (from a standard deck) with at least 3 kings? (enter an integer without commas)
There are 73,701 different 6-card hands (from a standard deck) with at least 3 kings.
To calculate the number of 6-card hands with at least 3 K's, the problem can be divided into:
Case 1:
Exactly 3 Kings
There are 4 ways to choose 3 kings to put in the hand, then there are 48 cards left to choose the remaining 3 cards (because we used 3 cards in a 52-card deck). Therefore, the number of 6-card hands with exactly 3 kings is:
4 * (48 choose 3) = 4 * 17,296 = 69,184
Case 2:
Exactly 4 Kings
There are 4 ways to choose 4 kings to put in the hand, then there are 48 cards left to choose the remaining 2 cards. Therefore, the number of 6-card hands with exactly 4 kings is:
4 * (48 choose 2) = 4 * 1.128 = 4.512
Case 3:
Exactly 5 kings
There are 4 ways to choose the 5 kings in the hand, then there is only one card left to choose from (because we used 5 of the 52 cards in the deck of cards). Therefore, the number of 6-card hands with exactly 5 kings is:
4*1=4
Case 4:
6 cards are king
There is only one way to choose all 6 cards as king.
Therefore, the total number of 6-card hands with at least 3 kings is:
69,184 + 4,512 + 4 + 1 = 73.701
So there are 73,701 different 6-card hands with at least 3 kings.
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suppose that 30% of new yorkers own a dog, 25% of new yorkers own a cat and 15% of new yorkers own a cat given they own a dog. a new yorker is chosen at random and reported to own a cat. what is the probability they also own a dog?
The probability that a New Yorker who possesses a cat also owns a dog is 0.103. if a new yorker is chosen at random.
New Yorkers own a dog (D) = 30%
New Yorkers own a cat (C) = 25%
New Yorkers own a cat and dog = 15%
This problem can be calculated using Bayes' theorem, which notes that the possibility of an event A given event B is equal to the possibility of event B given A times the probability of A, divided by the probability of event B.
P(D) = 0.30
P(C) = 0.25
P(C|D) = 0.15
Using Bayes' theorem:
P(D|C) = P(C|D) * P(D) / P(C)
P(C) = [tex]P(C|D) * P(D) + P(C|not D) * P(not D)[/tex]
P(C) = [tex]P(C|D) * P(D) + P(C) * P(not D)[/tex]
P(C) =[tex]P(C|D) * P(D) / (1 - P(D) * P(C|not D))[/tex]
Now we can substitute these values into the Bayes' theorem formula:
P(D|C) = P(C|D) * P(D) / P(C)
P(D|C) = [tex]0.15 * 0.3 / (P(C|D) * P(D) + P(C) * P(not D))[/tex]
P(D|C) = [tex]0.15 * 0.3 / (0.15 * 0.3 + P(C) * 0.7)[/tex]
P(C) = [tex]0.15 * 0.3 / (1 - 0.3 * 0.25)[/tex]
P(C) = 0.16
P(D|C) = [tex]0.15 * 0.3 / (0.15 * 0.3 + 0.16 * 0.7)[/tex]
P(D|C) = 0.103
Therefore, we can conclude that the probability that a New Yorker who owns a cat also owns a dog is 0.103.
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