Answer:
I speak englush
Step-by-step explanation:
I speak English
If John mows 11. 5 meters of lawn from east to west in 7. 1 seconds, what is the velocity of the lawnmower?
The velocity is 1.62 meters per second to the west.
What is the velocity of the lawnmower?We know that John mows 11.5 meters lan from east to west in 7.1 seconds.
Then we know that.
distance = 11.5 meters
time = 7.1 seconds.
To get the velocity, we just need to take the quotient between the distance and the time (and we need to clarifiy the direction), so we will get:
Velocity = distance/time
velocity = 11.5 meters/7.1 seconds
velocity = 1.62 meters per second to the west.
That is the velocity of the lawnmower.
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In two factor ANOVA, an F ratio is calculated for each different
sum of squares.
mean square.
factor.
null hypothesis.
In two factor ANOVA, an F ratio is calculated for each different sum of squares.
Specifically, the F ratio is obtained by dividing the mean square for a given factor or interaction by the mean square for error in two factor ANOVA. The sum of squares refers to the total variability that can be attributed to a particular factor or interaction, while the mean square is the sum of squares divided by its degrees of freedom. The F ratio is used to test the null hypothesis that the means of the different groups or levels within a factor are equal, and a significant F ratio indicates that there is evidence of a difference between at least two means.
ANOVA (Analysis of Variance) is a statistical method used to determine whether there are any significant differences between the means of three or more groups of data. ANOVA tests the null hypothesis that there is no difference between the means of the groups, based on the variance within and between the groups. It is often used in experimental research and can help identify factors that may be contributing to observed differences in data.
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The sampling distribution of the quantity: (n-1)s^2 / sigma^2 A. a t distribution B. a normal distribution C. an F distribution D. a chi-square distribution
That the sampling distribution of the quantity (n-1)s^2 / sigma^2 is a chi-square distribution.
When we have a sample of size n from a normal population with unknown variance sigma^2, we use the sample variance s^2 as an estimator for the population variance. However, the sample variance s^2 tends to underestimate the population variance sigma^2. To correct for this bias, we use (n-1)s^2 instead of ns^2 as an estimator for sigma^2.
The quantity [tex]\frac{(n-1)s^2}{sigma^2}[/tex] is called the sample variance ratio or the mean square ratio. It measures the ratio of the sample variance to the population variance. It is used in hypothesis testing and confidence interval construction for the population variance.
The distribution of the sample variance ratio is a chi-square distribution with (n-1) degrees of freedom. This means that if we take many random samples of size n from a normal population with unknown variance sigma^2 and calculate the sample variance ratio for each sample, the distribution of these ratios will follow a chi-square distribution with (n-1) degrees of freedom.
Therefore, we can conclude that the sampling distribution of the quantity (n-1)s^2 / sigma^2 is a chi-square distribution.
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Thus, the sampling distribution of (n-1)s^2 / sigma^2 is a chi-square distribution with n-1 degrees of freedom, assuming a normal population distribution.
The sampling distribution of the quantity (n-1)s^2 / sigma^2 is a chi-square distribution.
This is because the formula for the sample variance (s^2) involves subtracting the mean from each observation, squaring those deviations, and then summing them up. The resulting sum of squares follows a chi-square distribution with n-1 degrees of freedom. Dividing this sum of squares by sigma^2, the population variance, yields the quantity (n-1)s^2 / sigma^2. Since this is just a scaled version of the chi-square distribution, it also follows a chi-square distribution with n-1 degrees of freedom. It's important to note that this result assumes that the underlying population follows a normal distribution. If the population distribution is non-normal, the sampling distribution of (n-1)s^2 / sigma^2 may not follow a chi-square distribution.In such cases, alternative methods like bootstrapping or permutation tests may be used to estimate the variance.Know more about the chi-square distribution
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64% of U. S. Adults have very little confidence in newspapers you randomly select 10 U. S. Adults. Find the probability that the number of U. S. Adults who have very little confidence in news papers is (a) exactly five , (b) at least six, and (c) less than four
To solve this problem, we can use the binomial probability formula. The binomial distribution is applicable here because we have a fixed number of trials (selecting 10 U.S. adults) and each trial has two possible outcomes (having very little confidence or not having very little confidence in newspapers).
The formula for the probability of obtaining exactly 'k' successes in 'n' trials, where the probability of success is 'p', is:
[tex]P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)[/tex]
where C(n, k) represents the number of combinations of 'n' items taken 'k' at a time.
(a) To find the probability of exactly five U.S. adults having very little confidence in newspapers, we substitute the values into the formula:
[tex]P(X = 5) = C(10, 5) * (0.64)^5 * (1 - 0.64)^(10 - 5)[/tex]
Calculating this expression will give us the probability.
(b) To find the probability of at least six U.S. adults having very little confidence in newspapers, we need to calculate the sum of probabilities for six, seven, eight, nine, and ten successes:
P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
(c) To find the probability of less than four U.S. adults having very little confidence in newspapers, we need to calculate the sum of probabilities for zero, one, two, and three successes:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Using the binomial probability formula and the appropriate combinations, we can calculate these probabilities.
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How many different 5-letter symbols can be formed from the word YOURSELF if the symbol must begin with a consonant and ends with vowel?
There are 24 different 5-letter symbols that can be formed from the word "YOURSELF" if the symbol must begin with a consonant and end with a vowel.
To determine the number of different 5-letter symbols that can be formed, we need to consider the available choices for the first and fifth positions. The word "YOURSELF" has seven letters, out of which four are consonants (Y, R, S, and L) and three are vowels (O, U, and E).
Since the symbol must begin with a consonant, there are four choices for the first position. Similarly, since the symbol must end with a vowel, there are three choices for the fifth position.
For the remaining three positions (2nd, 3rd, and 4th), we can use any letter from the remaining six letters of the word.
Therefore, the total number of different 5-letter symbols that can be formed is calculated by multiplying the number of choices for each position: 4 choices for the first position, 6 choices for the second, third, and fourth positions (since we have six remaining letters), and 3 choices for the fifth position.
Thus, the total number of different 5-letter symbols is 4 * 6 * 6 * 6 * 3 = 24 * 36 = 864.
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Sted Overall in GCSE Mathematics (GCSE Maths FT Thu)
SER
**• rr:
Calculator Question
(0/3 Points)
Karim buys 200 tiles.
The tiles are sold in boxes.
There are 25 tiles in each box.
Each box of tiles costs £9. 75
Work out the total cost of the boxes of tiles Karim buys.
the total cost of the boxes of tiles Karim buys is £78.
To calculate the total cost of the boxes of tiles Karim buys, we need to multiply the number of boxes by the cost per box.
Given that there are 25 tiles in each box and Karim buys 200 tiles, we can determine the number of boxes as follows:
Number of boxes = Total number of tiles / Tiles per box
Number of boxes = 200 tiles / 25 tiles per box
Number of boxes = 8 boxes
Next, we multiply the number of boxes by the cost per box to find the total cost:
Total cost = Number of boxes * Cost per box
Total cost = 8 boxes * £9.75 per box
Total cost = £78
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Leo bought 3. 5lbs of strawberries that cost $4. 20. How many pounds could Leo buy with the same amount of money if the strawberries cost 2. 80 per pound
Leo could buy 1.5 pounds of strawberries if they cost $2.80 per pound.
How many pounds could Leo buy with the same amount of moneyFrom the question, we have the following parameters that can be used in our computation:
3. 5lbs of strawberries that cost $4.20.
This means that
Cost = $4.20
Pounds = 3.5
For a unit rate of 2.8 we have
Pounds = 4.20/2.8
Evaluate
Pounds = 1.5
Hence, Leo could buy 1.5 pounds of strawberries if they cost $2.80 per pound.
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let |a| 5 30. how many left cosets of ka4 l in kal are there? list them.
There are 6 left cosets of Ka4l in Kal.
How many left cosets of Ka4l are there in Kal?In abstract algebra, a left coset is a set formed by multiplying a fixed element on the left with each element of a subgroup. In this case, we have the subgroup Ka4l within the group Kal. The given condition states that |a| ≤ 5 ≤ 30, which means the element 'a' can take values from 1 to 5.
To determine the left cosets, we need to multiply each element of the subgroup Ka4l by the elements in Kal. The left cosets are essentially distinct sets of elements obtained by multiplying each element of Ka4l by all the elements of Kal.
The subgroup Ka4l consists of all elements of Kal that can be expressed as ka4l, where k is an element of Kal. Multiplying each element of Ka4l by elements of Kal, we obtain the following left cosets:
1. {ka4l : k ∈ Kal}
2. {2a4l : k ∈ Kal}
3. {3a4l : k ∈ Kal}
4. {4a4l : k ∈ Kal}
5. {5a4l : k ∈ Kal}
6. {6a4l : k ∈ Kal}
These are the six left cosets of Ka4l in Kal.
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Verify that (0, 0) and (10/3,0) are critical points of the following function: f(x, y) = 3x ^ 2 * y + 2x * y ^ 2 - 10xy - 8y ^ 2
Classify these given critical points into relative maximum, relative minimum or saddle
points.
The points (0, 0) and (10/3, 0) are critical points of the function f(x, y) = 3x^2 * y + 2x * y^2 - 10xy - 8y^2. The point (0, 0) is a saddle point, while the point (10/3, 0) is a relative minimum.
To determine the critical points, we need to find the values of x and y where the partial derivatives of the function f(x, y) with respect to x and y are both equal to zero.
Taking the partial derivative with respect to x, we have:
∂f/∂x = 6xy + 2y^2 - 10y
Taking the partial derivative with respect to y, we have:
∂f/∂y = 3x^2 + 4xy - 10x - 16y
Setting both partial derivatives equal to zero and solving, we find two critical points: (0, 0) and (10/3, 0).
To classify these critical points, we can use the second derivative test or evaluate the Hessian matrix. However, in this case, evaluating the Hessian matrix is not necessary. By observing the terms of the function, we can determine that the point (0, 0) is a saddle point because it changes sign when crossing the axes.
For the point (10/3, 0), we can evaluate the function at nearby points to determine its nature.
By plugging in values slightly greater and slightly smaller than 10/3 for x, we find that f(x, y) is positive for x slightly greater than 10/3 and negative for x slightly smaller than 10/3. Therefore, (10/3, 0) is a relative minimum.
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What is the volume for the following shape? Round your answer to the nearest tenth.
Answer:
418.7 cm³-------------------
Find the volume of the given cone:
V = πr²h/3Substitute to get:
V = 3.14*4²*25/3V = 418.7 cm³ (rounded)factor the gcf out of 6x^2+10
Retchen made a paper cone to hold a gift for a friend. The paper cone was 12 inches high and had a radius of 4 inches. Find the volume of the paper cone to the nearest tenth. Use 3. 14 for π. The volume of the paper cone is about
1. Which circle does the point (-1,1) lie on?
O (X2)2 + (y+6)2 - 25
0 (x-5)2 + (y+2)2 = 25
0 (x2)2 + (y-2)2 = 25
0 (x-2)2 + (y-5)2 = 25
The given options can be represented in the following general form:
Circle with center (h, k) and radius r is expressed in the form
(x - h)^2 + (y - k)^2 = r^2.
Therefore, the option with the equation (x + 2)^2 + (y - 5)^2 = 25 has center (-2, 5) and radius of 5.
Let us plug in the point (-1, 1) in the equation:
(-1 + 2)^2 + (1 - 5)^2 = 25(1)^2 + (-4)^2 = 25.
Thus, the point (-1, 1) does not lie on the circle
(x + 2)^2 + (y - 5)^2 = 25.
In conclusion, the point (-1, 1) does not lie on the circle
(x + 2)^2 + (y - 5)^2 = 25.
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compute t2(x) at x=0.6 for y=ex and use a calculator to compute the error |ex−t2(x)| at x=−1.5.
t2(0.6) = 0.6² = 0.36. Using a calculator, the error |ex − t2(x)| at x = -1.5 is approximately 2.352.
What are the values of t2(0.6) and the error |ex − t2(x)| at x = -1.5?To compute t2(0.6), we substitute x = 0.6 into the expression t2(x) = x², resulting in t2(0.6) = 0.6² = 0.36.
To determine the error |ex − t2(x)| at x = -1.5, we need to evaluate ex and t2(x) at x = -1.5. Using a calculator, we find that ex ≈ 4.48169 and t2(-1.5) = (-1.5)² = 2.25. Therefore, the error is calculated as |4.48169 - 2.25| ≈ 2.23169.
In summary, t2(0.6) is equal to 0.36, while the error |ex − t2(x)| at x = -1.5 is approximately 2.352.
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Which function best models the data?
Time, t (s) 0 0. 5 1. 0 1. 5 2. 0
Height, h (m) 3. 0 6. 8 8. 2 7. 0 3. 3
A. H(t) = −15. 9t^2 + 2. 99t + 10. 22
B. h(t) = −16. 1t^2 + 10. 22t + 2. 99
C. H(t) = −5. 03t^2 + 10. 22t + 2. 99
D. h(t) = −5. 03t^2 + 2. 99t + 10. 22
The quadratic term ([tex]-5.03t^2[/tex]) captures the curvature of the data, henceThe function that best models the given data is option C: [tex]H(t) = -5.03t^2 + 10.22t + 2.99[/tex].
To determine which function best models the data, we can compare the given data points to the equations provided.
The given data consists of time, t (in seconds), and height, h (in meters). By observing the patterns in the data, we can determine the appropriate equation.
Comparing the data points with the equations, we find that option C, [tex]H(t) = -5.03t^2 + 10.22t + 2.99[/tex], best fits the given data. This equation represents a quadratic function, which matches the curved pattern of the data.
In option C, the coefficients and exponents of the equation closely correspond to the given data points. The quadratic term[tex](-5.03t^2)[/tex] captures the curvature of the data, and the linear terms [tex](10.22t + 2.99)[/tex]account for the overall trend of the data points.
Therefore, the best function that models the given data is C: [tex]H(t) = -5.03t^2 + 10.22t + 2.99.[/tex]
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Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of r given the number of pairs of data n and the significance level alpha n = 11, a = 0.01
Thus, For n = 11 and alpha = 0.01, the critical values of r are approximately -0.869 and 0.869. These values are the boundaries for determining whether the correlation is significant.
To determine if there is sufficient evidence to support a claim of a linear correlation between two variables, you can perform a hypothesis test using the correlation coefficient, r. The critical values of r will help you decide if the correlation is significant or not.
For a given number of pairs of data (n) and a significance level (alpha), you can find the critical values of r using a table of critical values for the Pearson correlation coefficient or an online calculator.
In your case, you have n = 11 pairs of data and a significance level of alpha = 0.01. Using a table or calculator, you can find the critical values for a two-tailed test.
For n = 11 and alpha = 0.01, the critical values of r are approximately -0.869 and 0.869. These values are the boundaries for determining whether the correlation is significant.
If the calculated value of r falls between these critical values (-0.869 and 0.869), you would fail to reject the null hypothesis, meaning there is insufficient evidence to support a claim of a linear correlation between the two variables.
However, if the calculated value of r is less than -0.869 or greater than 0.869, you would reject the null hypothesis, indicating sufficient evidence to support the claim of a linear correlation at the 0.01 significance level.
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What is the radius of convergence of a power series function?
The radius of convergence of a power series function is a positive real number R that determines the interval of values for which the power series converges.
It represents the distance from the center of the power series expansion, within which the series converges. The radius of convergence is determined by the properties of the coefficients in the power series. Specifically, it is defined as the reciprocal of the limit superior of the absolute values of the coefficients. Mathematically, if we have a power series function of the form:
f(x) = ∑(n=0 to ∞) aₙ(x - c)ⁿ
where aₙ represents the coefficients and c is the center of the series, then the radius of convergence R is given by:
R = 1 / lim sup |aₙ|^(1/n)
The power series converges for all values of x within the interval (c - R, c + R). If |x - c| > R, the series diverges.
It's important to note that the radius of convergence can be zero, indicating that the power series only converges at the center point (x = c), or it can be infinite, indicating that the series converges for all values of x.
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S • 41. If US$ 1 is equivalent to $ 47.50, the value of US$7 in Jamaican currency is?
Answer:
your anwser is 1085
Step-by-step explanation:
let ~u and ~v be vectors in three dimensional space. if ~u ×~v = ~0, then ~u = ~0 or ~v = ~0. state if this is true or false. explain why.
The statement is true because if the cross product of two vectors ~u and ~v in three-dimensional space is equal to the zero vector ~0, then it implies that either ~u or ~v is equal to the zero vector ~0.
The cross product ~u × ~v produces a vector that is perpendicular (orthogonal) to both ~u and ~v. If the resulting cross product is the zero vector ~0, it means that ~u and ~v are either parallel or collinear.
If ~u and ~v are parallel or collinear, it implies that they are scalar multiples of each other. In this case, one of the vectors can be expressed as a scaled version of the other. Consequently, either ~u or ~v can be the zero vector ~0.
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Plssssssss help pls need thisss
The expression that shows the total area of the shape is 4s²
What is area of shape?The area of a shape is the space occupied by the boundary of a plane figures like circles, rectangles, and triangles.
The shape above consist of 4 equal squares, each sides of the square is 's'. This means that the area of one square will be area of the remaining 3 squares.
Area of a square is expressed as;
A = l²
where l is the side length
area of one square = s × s
= s²
For 4 squares now, the total area will be
s² + s² + s² + s²
= 4s²
Therefore the total area of the shape is 4s²
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consider the function f(x)={xif x<11xif x≥1 evaluate the definite integral. ∫08f(x)dx
To evaluate the definite integral [tex]\int\limit {0^{8} fx} \, dx[/tex], we first need to identify the values of the function f(x) in the given interval [0, 8].
Since 0 < 1, we know that f(0) = 0. Similarly, since 8 < 11, we know that f(8) = 8.
Next, we need to evaluate the integral of f(x) over the interval [0, 8]. Since the function f(x) is defined piecewise, we need to split the interval into two parts: [0, 1) and [1, 8].
Over the interval [0, 1), the function f(x) is equal to 0. Therefore, the integral of f(x) over this interval is equal to 0.
Over the interval [1, 8], the function f(x) is equal to x. Therefore, the integral of f(x) over this interval is equal to:
[tex]\int\limits {1^{8} x} \, dx=\int\limit \frac{x^{2} }{2}} 1^{8} = \frac{8^{2} }{2} -\frac{1^{2} }{2}=28[/tex]
So, the answer to the question is 28.
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Find the area of the rectangle ABCD with vertices A(-4, 4), B(1, 4), C(-4, 1) and D(1,1).
The area of the rectangle ABCD is approximately 29.15 square units.
To find the area of the rectangle ABCD, we can use the formula for the area of a rectangle, which is given by the product of its length and width.
Let's first find the length and width of the rectangle using the coordinates of its vertices.
Length AB = distance between points A and B
= √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(1 - (-4))² + (4 - 4)²]
= √[5² + 0²]
= √25
= 5
Width BC = distance between points B and C
= √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(-4 - 1)² + (1 - 4)²]
= √[(-5)² + (-3)²]
= √[25 + 9]
= √34
Now that we have the length and width, we can calculate the area of the rectangle.
Area = Length × Width
= 5 × √34
≈ 5 × 5.83
≈ 29.15 square units
Therefore, the area of the rectangle ABCD is approximately 29.15 square units.
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a type of diagram that is used to graphically show the relationship between two numerical variables.
The type of diagram used to graphically show the relationship between two numerical variables is called a scatter plot.
A scatter plot is a visual representation of data points plotted on a graph, with one variable represented on the x-axis and the other variable represented on the y-axis.
Each data point on the plot corresponds to a pair of values from the two variables being analyzed. The position of each point on the graph indicates the values of both variables, allowing us to examine the relationship between them.
The main purpose of a scatter plot is to visualize the correlation or relationship between the two variables. The pattern formed by the data points on the plot can indicate the direction, strength, and nature of the relationship.
For example, if the points on the scatter plot tend to form a linear pattern, it suggests a linear relationship between the variables. On the other hand, if the points are scattered randomly with no clear pattern, it indicates a weak or no relationship between the variables.
Scatter plots are commonly used in various fields, including statistics, data analysis, and scientific research. They provide a visual way to explore and interpret relationships between variables, identify outliers, detect trends, and assess the strength and direction of associations.
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One trampoline has a diameter of 12 feet. A larger trampoline has a diameter of 14 feet. How much greater is the area of the larger trampoline? Use 3.14 for pi and round your answer to the nearest hundredths.
The area of the larger trampoline is 40.82 ft² greater than the area of the smaller trampoline.
How to calculate the area of a circle?The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:
A = πr²
The radius of the smaller trampoline is given as follows:
6 feet (half the diameter).
Hence the area is given as follows:
A = 3.14 x 6²
A = 113.04 ft².
The radius of the larger trampoline is given as follows:
7 ft.
Hence the area is given as follows:
A = 3.14 x 7²
A = 153.86 ft².
Then the difference of the areas is given as follows:
153.86 - 113.04 = 40.82 ft².
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The Mosteller formula for approximating the surface area S, in square meters (m2), of a human is given by the function below, where h is the person's height in centimeters and w is the person's weight in kilograms. According to this formula, if a person's weight drops 17%, by what percentage does his or her surface area change? Vhw S(h,w) = 60 Choose the correct answer below. A. It drops by approximately 40%. B. It drops by approximately 20%. C. It drops by approximately 30%. OD. It drops by approximately 10%.
The surface area is changed by around 40% which means It drops by approximately 40%.
Option A is the correct answer.
We have,
To find the percentage change in surface area, we need to calculate the new surface area after the weight drop and then find the percentage difference.
Let the original weight be w, and the new weight after the 17% drop be w(new) = w - 0.17w = 0.83w.
The original surface area.
S(h, w) = √(hw) / 60.
The new surface area.
S(h, w_new) = √(h x 0.83w) / 60.
To find the percentage change, we calculate the difference between the two surface areas and divide it by the original surface area, then multiply by 100:
Percentage Change
= [(S(h, w) - S(h, w(new))) / S(h, w)] x 100
Now let's plug in the formula for surface area:
Percentage Change
= [((√(hw) / 60) - (√(h * 0.83w) / 60)) / (√(hw) / 60)] * 100
= [(√(hw) - √(h * 0.83w)) / √(hw)] * 100
= [0.398w / √(hw)] * 100
= 39.8%
= 40%
Thus,
The surface area is changed by around 40% which means It drops by approximately 40% which is option A.
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The complete question:
The Mosteller formula for approximating the surface area S, in square meters (m²), of a human is given by the function below, where h is the person's height in centimeters and w is the person's weight in kilograms.
S(h, w) = √(hw) / 60
According to this formula, if a person's weight drops 17%, by what percentage does his or her surface area change?
Choose the correct answer below.
A. It drops by approximately 40%.
B. It drops by approximately 20%.
C. It drops by approximately 30%.
D. It drops by approximately 10%.
Write the repeating decimal as a fraction. .1872 72 is a repeating decimal.
The fraction representation of the repeating decimal .1872 72 is 18727/99900.
To express the repeating decimal .1872 72 as a fraction, we can follow these steps:
Let x = .187272...
Step 1: Multiply both sides of the equation by a power of 10 to shift the repeating part to the left of the decimal point. Since there are two digits in the repeating part, we can multiply by 100:
100x = 18.727272...
Step 2: Subtract the original equation from the multiplied equation to eliminate the repeating part:
100x - x = 18.727272... - 0.187272...
99x = 18.54
Step 3: Divide both sides by 99 to isolate x:
x = 18.54 / 99
Simplifying the fraction:
x = 927 / 4950
Therefore, the fraction representation of the repeating decimal .1872 72 is 927/4950.
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In ΔMNO, the measure of ∠O=90°, the measure of ∠M=13°, and OM = 9. 6 feet. Find the length of MN
In a right triangle, the right angle is marked as 90°. Here, ∠O is marked as 90°, indicating that the triangle is a right triangle.
Moreover, the length of OM is given as 9.6 feet. The formula used to find the length of the hypotenuse is Pythagoras theorem. The formula is given as c² = a² + b². In a right triangle, the hypotenuse is marked as c, and a and b are the other two sides.
Let's use Pythagoras theorem to find the length of the hypotenuse, MN. MN is the hypotenuse.c² = a² + b²c² = 9.6² + 13²c² = 92.16 + 169c² = 261.16The square root of 261.16 is 16.16. Therefore, the length of MN is 16.16 feet. This is the required solution. In conclusion, using Pythagoras theorem, we can find the length of the hypotenuse of a right triangle if the lengths of the other two sides are given.
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Give a practical interpretation in words of the function
1) k(g(t)), where L=k(H) is the length of a steel bar at temperature H and H=g(t) is temperature at time t
2) t(f(H)), where t(v) is the time of a trip at velocity v, and v=f(H) is velocity at temperature H
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Find a simplified formula for the difference quotient --- (f(x+h)-f(x))/h
3) f(x)=x^2 +x
4) f(x)=sqrtx
5) f(x)= 1/x
Function k(g(t)) is used to find length of steel bar at any given time based on the temperature.
Function t(f(H)) is help us to find time taken to travel a certain distance at any given temperature based on velocity.
(f(x + h) - f(x)) / h = 2x + h + 1
(f(x+h) - f(x)) / h = 1 / (√(x+h) +√(x))
(f(x+h) - f(x)) / h = -1 / (x(x+h))
The function k(g(t)) gives the length of a steel bar L, at a certain temperature H, where H is a function of time, g(t).
This means that the length of the steel bar is dependent on the temperature of the bar, which in turn depends on the time.
The function k(g(t)) is used to determine the length of the bar at any given time based on the temperature.
The function t(f(H)) gives the time it takes to travel a certain distance at a given velocity v, where v is a function of temperature H.
The time of the trip is dependent on the velocity of travel, which in turn depends on the temperature.
The function t(f(H)) is used to determine time it takes to travel a certain distance at any given temperature based on the velocity.
The difference quotient for f(x) = x² + x is,
(f(x+h) - f(x)) / h = [(x+h)² + (x+h) - (x² + x)] / h
Simplifying this expression, we get,
⇒ (f(x+h) - f(x)) / h = [(x² + 2xh + h² + x + h) - (x² + x)] / h
⇒ (f(x+h) - f(x)) / h = (2xh + h² + h) / h
⇒ (f(x+h) - f(x)) / h = 2x + h + 1
The difference quotient for f(x) = √(x) is,
(f(x+h) - f(x)) / h = (√(x+h) - √(x)) / h
Multiplying the numerator and denominator by the conjugate of the numerator, we get,
(f(x+h) - f(x)) / h = [(√(x+h) - √(x)) × (√(x+h) + √(x))] / [h × (sqrt(x+h) + sqrt(x))]
⇒ (f(x+h) - f(x)) / h = (x+h - x) / [h × (√(x+h) + √(x))]
⇒ (f(x+h) - f(x)) / h = 1 / (√(x+h) + √(x))
The difference quotient for f(x) = 1/x is,
⇒ (f(x+h) - f(x)) / h = (1 / (x+h) - 1 / x) / h
Multiplying the numerator and denominator by x(x+h), we get,
⇒ (f(x+h) - f(x)) / h = [(x - (x+h)) / (x(x+h))] / h
⇒ (f(x+h) - f(x)) / h = (-h / (x(x+h))) / h
⇒ (f(x+h) - f(x)) / h = -1 / (x(x+h))
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The area of a square is increasing at a rate of 80 centimeters squared per second. Find the rate of change of the side of the square when it is 8 centimeters. The rate of change of the side is Number cm/sec. In a few sentences, please explain how you got your answer.
The rate of change of the side length when the area is 8 cm² is 5 cm/sec.
The area of a square is given by the formula A = s², where A is the area and s is the length of one side of the square. We are given that the area is increasing at a rate of 80 cm²/sec. Using implicit differentiation, we can find the rate of change of the side length when the area is 8 cm².
dA/dt = 2s(ds/dt)
Substituting in the given values, we get:
80 = 2(8)(ds/dt)
ds/dt = 5 cm/sec
Therefore, the rate of change of the side length when the area is 8 cm² is 5 cm/sec.
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Answer true or false:A linear programming problem may have more than one optimal solution.
True. A linear programming problem may indeed have more than one optimal solution. Linear programming is a method used to determine the best outcome or solution from a given set of resources and constraints.
It involves optimizing a linear objective function, which represents the goal of the problem, subject to a set of linear inequality or equality constraints. In some cases, a linear programming problem can have multiple optimal solutions, which means that there is more than one solution that satisfies the constraints and provides the best possible value for the objective function. This can occur when the feasible region, which is the set of all possible solutions that satisfy the constraints, has more than one point that lies on the same level curve of the objective function. When a problem has multiple optimal solutions, it is said to have degeneracy. Degeneracy can arise due to various reasons, such as redundant constraints or parallel objective function lines. In these situations, any of the optimal solutions can be chosen, as they all yield the same optimal value for the objective function. It is true that a linear programming problem may have more than one optimal solution, and understanding the reasons for degeneracy can help in identifying and selecting the most suitable solution for a specific problem.
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