The spot on the shoreline is moving at a rate of about 1.764 feet per minute when it is 13 feet from the point on the shoreline closest to the lighthouse.
What is differentiation?
Differentiation is a mathematical operation that is used to find the rate at which a function changes. More specifically, it is the process of finding the derivative of a function.
The derivative of a function at a given point is a measure of how quickly the function is changing at that point. It gives the slope of the tangent line to the graph of the function at that point. The derivative can be thought of as the instantaneous rate of change of the function at that point.
Let's call the point on the shoreline closest to the lighthouse "P". We know that the distance from the spotlight to P is 130 feet, and we want to find the rate at which the distance from the spotlight to P changes when the spotlight is 13 feet from P.
To do this, we can use the chain rule to differentiate the distance formula with respect to time. Let's call the distance between the spotlight and P "d". Then:
d²= 130² + x²
Taking the derivative of both sides with respect to time gives:
2d * dd/dt = 0 + 2x * dx/dt
We can solve for dd/dt by plugging in the values we know:
130² + 13² = d²
d = [tex]\sqrt{(130^2 + 13^2)}[/tex] = 130.325 ft
2(130.325) * dd/dt = 2(13) * dx/dt
dd/dt = (13/130.325) * dx/dt
We know that the spotlight revolves at a rate of 281 rad/min, or 281/2π ≈ 44.7 revolutions per minute. Each revolution of the spotlight covers a distance of 2π * 130 feet, so its speed is:
(2π * 130 ft/rev) * (44.7 rev/min) = 18410.8 ft/min
To find dx/dt when x = 13, we need to find the angular velocity of the spotlight at that point. The spotlight makes one full revolution every 60/14 ≈ 4.29 seconds, so its angular velocity is:
2π radians/rev ÷ 4.29 s/rev = 1.47 radians/s
At any given moment, the angle between the spotlight and the line connecting the lighthouse and P is equal to the arctangent of x/130. When x = 13, this angle is:
arctan(13/130) ≈ 5.71°
The rate at which the angle is changing is equal to the angular velocity of the spotlight, so we can use the formula for the derivative of the arctangent to find dx/dt:
dx/dt = 130 * tan(5.71°) * (1.47 radians/s)
dx/dt ≈ 17.602 ft/min
Finally, we can substitute this value into the expression we found for dd/dt:
dd/dt = (13/130.325) * 17.602
dd/dt ≈ 1.764 ft/min
So the spot on the shoreline is moving at a rate of about 1.764 feet per minute when it is 13 feet from the point on the shoreline closest to the
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x f 20-24 2 15-19 5 10-14 4 5-9 1 for the distribution in table 2-3, what was the highest score obtained in this group of 12 scores? group of answer choices
The highest score obtained in this group of 12 scores is 19
What is the distribution in statistics?
In statistics, a distribution refers to the way in which a variable is spread out or distributed across different values. A distribution describes the probability of different outcomes for a variable, and can be visualized using graphs such as histograms, box plots, and probability density functions.
The distribution in Table 2-3 shows frequency (f) for different score ranges (x). Without knowing the exact scoring system, we can't say for certain what the highest score obtained in this group of 12 scores is. However, we can determine the upper limit of the highest score range by multiplying the upper limit of the range by its frequency:
Upper limit of 20-24 range = 24
Frequency of 20-24 range = 2
Upper limit of 15-19 range = 19
Frequency of 15-19 range = 5
Upper limit of 10-14 range = 14
Frequency of 10-14 range = 4
Upper limit of 5-9 range = 9
Frequency of 5-9 range = 1
To find the highest score, we need to determine the upper limit of the score range that has the highest frequency. In this case, the highest frequency is 5, which corresponds to the range 15-19.
Hence, the highest score range is 15-19, and the highest score within that range is the upper limit of that range, which is 19. So, the highest score obtained in this group of 12 scores is 19 (assuming the scoring system is such that higher scores correspond to higher ranges).
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Forty-two percent of what number is 168? Responses
40
70.6
400
705.6
Answer:
Your answer is: C) 400
Step-by-step explanation:
please help with q26 and show working
Answer:
13 square units
Step-by-step explanation:
The composite figure can be decomposed into
2 semicircles each of diameter 3
1 rectangle of height 1 and length 5
1 square of side 1
Radius of each semicircle = 3/2 = 1.5 units
Area of a semicircle of radius r = 1/2 area of circle of radius r = 1/2 x π x r²
Two semicircles make a circle
So area of the two semicircles = 2 x (1/2 x π x r²) = πr² = π x 1.5²
= 2.25π = 7.07
= 7 square units rounded to nearest whole number
Area of the rectangle = width x length = 1 x 5 = 5 square units
Area of square of side 1 = 1 x 1 = 1 square unit
Total area of shaded area = 7 + 5 + 1 = 13 square units
Can you please find the area and perimeter
The area and perimeter of the figures are( 15a³b⁴ and 10a²b⁴ + 6ab²), (6m³n² and 4m² + 10mn²) and (8m³n and 4√[m(m+2y²)] + 8mn)
What is the area and perimeter of the figures1.
The area of the rectangle can be calculated using the formula;
A = l * w
A = 5a²b⁴ * 3ab²
A = 15a³b⁴
The perimeter of the rectangle is
P = 2(l + w)
P = 2[(5a²b⁴) + (3ab²)
P = 10a²b⁴ + 6ab²
2.
The area of a triangle is given by
A = 1/2 b * h
A = 0.5 * 4m² * 3mn²
A = 6m³n²
The perimeter of the triangle is calculated as
P = 4m² + 7mn² + 3mn²
P = 4m² + 10mn²
3.
The area of the triangle is given as;
A = 1/2 * b * h
A = 0.5 * 4mn * 4m²
A = 8m³n
The perimeter of the triangle is
Using Pythagoras theorem to find the hypothenuse side
The third side is 2√[m(m + 2n²)]
The perimeter of the triangle is = 4√[m(m+2y²)] + 8mn
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Answer: look at the image for the answer and solution
Step-by-step explanation:
A square lawn has an area of 1922 square feet. A sprinkler placed at the center of the lawn sprays a water in a circular pattern that just covers the lawn. What is the radius of the circle?
A. 43. 84 B. 21. 92 C. 62 D. 31
The radius of the circle such that it just covers the lawn is B. 21.92.
The circular pattern just covering the square lawn will have one thing in common. It is the length or width of square which will be equal to diameter of the circle. So, finding the same to calculate our answer.
Area of square = side²
Side = ✓area
Keep the values
Side = ✓1922
Side = 43.84 feet
The side of the square will be the diameter. We know diameter is double of radius. So,
radius = 43.84/2
Performing division
Radius = 21.92 feet
Hence, correct option is B. 21.92
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If you are obtaining a $195,000 loan and the bank charges 4.1%
plus $250 and 2½ points, what are the one-time charges?
Answer:
5125
Step-by-step explanation:
I was have it in my practice today
When a scientist conducted a genetics experiments with peas, one sample of offspring consisted of 933 peas, with 729 of them having red flowers. If we assume, as the scientist did, that under these circumstances, there is a 3/4 probability that a pea will have a red flower, we would expect that 699.75 (or about 700) of the peas would have red flowers, so the result of 729 peas with red flowers is more than expected.
. If the scientist's assumed probability is correct, find the probability of getting 729 or more peas with red flowers.
b. Is 729 peas with red flowers significantly high?
c. What do these results suggest about the scientist's assumption that 3/4 of peas will have red flowers
The solution to the statistics have been given below
How to solve for the probabilityThe probability = 3 / 4 = 0.75
n = 933
x≥ 729
np = 0.75 * 933
= 699.75
√933 * 0.75 * 0.25
= √174.9375
= 13.23
using the continuity correction we have
729 - 0.5
= 728.5 - 699.75 / 13.23
= 2.17
The p value is 0.015003.
there is a 1.5% chance of getting 729 peas with red flowers
b. given that Z is greater than 2, this can be said to be high
c. These results suggest that the scientist maybe wrong in his assumption
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You take an SRS of size 500 from the 37,000 students at Purdue University and measure individual’s heights. You then take an SRS of size 250 from the 4,400,000 adults in the state of Indiana and measure their heights. Assuming the standard deviation of individual heights in the two populations is the same, the standard
deviation of the sampling distribution of mean heights for the Indiana sample is
(a) larger than for the Purdue sample because the sample size is smaller.
(b) larger than for the Purdue sample because it uses a smaller fraction of the population.
(c) smaller than for the Purdue sample because the sample size is smaller.
(d) smaller than for the Purdue sample because it uses a smaller fraction of the population.
(e) impossible to determine because of sampling variability.
Larger than for the Purdue sample because the sample size is smaller. Therefore, the option A is the correct answer.
What is margin of error?Your results will vary from the actual population value by how many percentage points, as indicated by the margin of error. A 95% confidence interval with a 4% margin of error, for instance, indicates that your statistic will, 95% of the time, be within 4% of the true population figure.
Given sample size = 500
Total outcomes = 37,000
The margin of error in a 95% confidence
The margin of error is given by
Margin of error (statistic) = Critical value x Standard error of the sample.
So, according to the given condition margin of error in a 95% confidence is same as for the Purdue sample because both are samples of size 500, because it depends on sample size.
Therefore, the option A is the correct answer.
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college gpa and salary. do students with higher college grade point averages (gpas) earn more than those graduates with lower gpas (civicscience)? consider the college gpa and salary data (10 years after graduation) provided in the file gpasalary. a. develop a scatter diagram for these data with college gpa as the independent variable. what does the scatter diagram indicate about the relationship between the two variables? b. use these data to develop an estimated regression equation that can be used to predict annual salary 10 years after graduation given college gpa. c. at the .05 level of significance, does there appear to be a significant statistical relationship between the two variables?
The scatter diagram for the data with college GPA as the independent variable is illustrated as follows.
In this case, we have a dataset that includes the GPA and salary of graduates ten years after graduation. We can use GPA as the independent variable (or x-axis) and salary as the dependent variable (or y-axis). We can then plot each data point on the graph, where the x-coordinate represents the GPA and the y-coordinate represents the salary.
To create the scatter plot, we first need to organize the data into pairs of GPA and salary values. For example, the first pair is (2.22, 72,000), where 2.22 is the GPA and 72,000 is the salary. We can then plot this point on the graph by locating 2.22 on the x-axis and 72,000 on the y-axis.
We repeat this process for each data point, and then connect the points with a line or a curve. The resulting scatter plot shows us the overall pattern of the relationship between GPA and salary. If there is a positive relationship between the two variables, we would expect to see the points cluster around a line that slopes upwards from left to right.
Overall, the scatter plot provides a useful visual tool for exploring the relationship between college GPA and salary. While it suggests that there is a positive relationship between the two variables, it also reminds us that there are many other factors that contribute to earnings.
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Complete Question:
Do students with higher college grade point averages (GPs) earn more than those graduates with lower GPAs?
Consider the following hypothetical college GPA and salary data (10 years after graduation). GPA Salary (S)
72,000 2.22
48,000 2.29
72,000 2.57
64,000 2.59
86,000 2.77
98,000 3.12
133,000 3.35
130,000 2.85
157,000 3.66
162,000 3.68
Develop a scatter diagram for these data with college GPA as the independent variable
I'm stuck on this question and I need help
A triangle has sides with lengths 8, 15, and 17.
a. Verify this is a Pythagorean triple.
b. Approximate the acute angles in this triangle.
Hence, It is proved that, Pythagorean theorem, is verified.
What is Pythagorean theorem?Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.
here, we have,
A triangle has sides with lengths 8, 15, and 17.
so, here, 8^2 + 15^2
=64 +225
=289
=17^2
Hence, It is proved that, Pythagorean theorem, is verified.
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What is the slope of the line that passes
through the points (10, 3) and (7,0)?
Write your answer in simplest form.
Answer:
slope = 1
Step-by-step explanation:
[tex]m=\frac{0-3}{7-10} =\frac{-3}{-3} =1[/tex]
Hope this helps.
A score that is 3 points lower than the sample mean has a z-score of z=-0.25, and a score of X= 44 has a z-score of -0.75. What is the sample mean? O a. M=48 b. M=56 c. M=51 d. M=53
The sample mean is c) M=51.
To find the sample mean, we can use the formula for z-score:
z = (X - M) / s
where:
In this case, we have two equations with two unknowns:
-0.25 = (M - 3 - M) / s
-0.75 = (44 - M) / s
Multiplying both sides of the first equation by s gives us:
-0.25s = M - 3 - M
Simplifying gives us:
-0.25s = -3
Next, we can multiply both sides of the second equation by s:
-0.75s = 44 - M
Now we can substitute the value of s from the first equation into the second equation:
-0.75(-3 / 0.25) = 44 - M
Simplifying gives us:
9 = 44 - M
Finally, we can solve for M:
M = 44 - 9
M = 35
So the sample mean is 35.
However, this answer is not one of the options given. It's possible that there was a mistake in the original question or in the calculations. Double-checking the calculations and the original question can help determine the correct answer. In this case, the correct answer is c. M=51.
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∠A and ∠ B are complementary angles. If m ∠ A = ( 2 x + 5 ) and m ∠ B = ( 4 x + 19 ) , then find the measure of ∠ A
The measure of m∠A is 27.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
m∠A = (2x + 5) and m∠B = (4x + 19)
∠A and ∠ B are complementary angles.
So,
Solve for x.
(2x + 5) + (4x + 19) = 90
2x + 5 + 4x + 19 = 90
6x + 24 = 90
6x = 90 - 24
6x = 66
x = 11
Now,
m∠A.
= (2x + 5)
= 2 x 11 + 5
= 22 + 5
= 27
Thus,
m∠A = 27
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what is 10000 = 8000(1 + .04t) interest is simple 4%. how long would it take for 8000 to grow to 10000
The number of years to grow the amount from $8,000 to $10,000 will be 6.25 years.
What is simple interest?Let P be the principal, R be the rate of interest, and T be the time. Then the interest rate is given as,
The formula for interest is written as,
I = (PRT)/100
The equation is given below.
10000 = 8000(1 + .04t)
Where 't' represents the number of years. Then solve the equation for 't', then we have
10000 = 8000(1 + 0.04t)
1.25 = 1 + 0.04t
0.04t = 0.25
t = 0.25 / 0.04
t = 6.25 years
The number of years to grow the amount from $8,000 to $10,000 will be 6.25 years.
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Mari bought 6 packets of tomato seeds. Each packet contained 24 seeds. She planted 1 packet of the seeds, and 15 seeds sprouted.Which statement about the seeds in the remaining packets is best supported by this information?
Answer:
Mari bought 6 packages of tomato seeds and each package contains 24 seeds then
Seeds = (6) * (24) = 144.
Mari has a total of 144 seeds.
Mari planted a pack of seeds and has 15 germinated seeds.
144-24-15 = 105
Mari then has 105 seeds left.
Answer
C At least 100 but no more than 120 seeds will sprout
The statement about the seeds in the remaining packets is best supported by this information is option C: At least 100 but no more than 120 seeds will sprout
What is the seeds about?Mari bought 6 packages of tomato seeds and each package contains 24 seeds so:
Seeds = 6 x 24 = 144.
Note that Mari has a total of 144 seeds. In the question, Mari planted a pack of seeds and has 15 germinated seeds.
Hence:
144-24-15 = 105
Mari then has 105 seeds left.
Hence option c: At least 100 but no more than 120 seeds will sprout, is correct.
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See full text below
Mari bought 6 packets of tomato seeds. Each
packet contained 24 seeds. She planted 1 packet
of the seeds, and 15 seeds sprouted.
Which statement about the seeds in the remaining
packets is best supported by this information?
A No more than 50 sees will sprout
B Between 50 and 100 seeds will sprout
C At least 100 but no more than 120 seeds will
sprout
D All 120 seeds will sprout
factor out the greatest common factor from 6x^4 + 8x^3 PLEASE HELP
Answer: 2x^3(3x+4)
Step-by-step explanation: 2x3x^4+2^3xx^
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
x+y=-4
-x-y=4
The following system of equations has no solutions, the correct option is A.
How to find the solution to the given system of equation?For that , we will try solving it first using the method of substitution in which we express one variable in other variable's form and then you can substitute this value in other equation to get linear equation in one variable.
If there comes a = a situation for any a, then there are infinite solutions.
If there comes wrong equality, say for example, 3=2, then there are no solutions, else there is one unique solution to the given system of equations.
We have given that;
system of equations are
x+y=-4.....1
-x-y=4......2
Now,
Rearranging the equation 2
-4=x+y
Its equal to equation 1
Therefore, the system of equations will have no solution.
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What is the HCF of 1854 and 81
The highest common factor, HCF of 18, 54, and 81 is 9.
What is the highest common factor?The highest common factor is the highest factor or number that can divide two or more numbers evenly without a remainder.
The common factors of 18, 54, and 81 are 1, 2, 3, 6, and 9 because each of these numbers can divide the three numbers without a remainder.
Based on the common factors of 18, 54, and 81, we can identify that the highest factor is 9.
The highest common factor, HCF, is also described as the greatest common factor, GCF.
Thus, the HCF of 18, 54, and 81 is 9.
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Complete Question:What is the HCF of 18, 54, and 81?
Find the value of x.
34°
4x²
Answer:
give the diagram please
Oliver tries to cut a piece of metal 30 centimeters long the price of metal ends up being 29. 4 centimeters long what is the percent error
The percent error in the specified situation is 2%.
The actual length of the metal is 30 centimeters, and the measured length is 29.4 centimeters. The difference between the actual and measured lengths is:
30 - 29.4 = 0.6 centimeters
The absolute error is the magnitude of this difference:
|0.6| = 0.6 centimeters
To find the percent error, we divide the absolute error by the actual value and multiply by 100:
percent error = (absolute error / actual value) x 100%
= (0.6 / 30) x 100%
= 2%
Therefore, the percent error is 2%.
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Will has 7/8 gallon of cold water in the refrigerator. During the day, he drinks 3/4 gallon of the water.
How much cold water does he have left at the end of the day?
A. 1/8 gallon
B. 1/4 gallon
C. 1/2 gallon
D. 1 gallon
Answer: 1/8
Step-by-step explanation: 3/4 is the same as 6/8
7/8- 6/8 = 1/8
The cost for renting a car is $75 plus $0.30 for every mile driven. Which equation,
written in function notation, represents this situation where R represents the total
cost for the rental and d represents miles driven?
R(d) = 0.30d + 75
R = 75d + 0.30
R(d) = 75d + 0.30
R = 75.30
Answer:R(d)=0.30d+75
Step-by-step explanation:
Note: This equation, as requested, only shows the cost of
rental on a daily basis. In order to solve for more than 1
day the equation should be: Cost = .3r+ 100d or something
similar... with r representing miles driven, d representing
total number of days for the rental.
The number of students enrolled at a college 18,000 and grows 5% each year. What is the initial amount
Answer:
900
Step-by-step explanation:
5% of 18,000
A Girl Scout is selling cookies to raise money for her troop. She is given a certain number of cookies to sell and is tracking the number of boxes remaining each
day. The table shows the relationship between the number of days she has been selling cookies and the total number of boxes remaining.
Girl Scout Cookie Fundraiser
Time (in days) 1 3 4 7
Number of Boxes 25 21 19 13
Which of the following graphs shows the relationship given in the table?
The corresponding graph for the given data is a coordinate plane with the x axis labeled time in days and the y axis labeled number of boxes, with a line that passes through the points (0, 27), (1,25), (3,21), (4,19) and (7,13)
What is a linear function?
A linear function is a polynomial function whose degree is utmost zero or one. It is represented as a straight line in the graph.
A Girl scout is selling cookies to raise money for her troop.
She is given a certain number of cookies to sell and is tracking the number of boxes remaining each day.
This result is shown in the table:
Time(in days) Number of boxes
1 25
3 21
4 19
7 13
The graph for the given data is attached below.
This forms a straight line.
By critically observing the graph and relationship between the time and number of boxes, we determine a function for the graph.
That is y=-2x+27.
Hence, the corresponding graph for the given data is a coordinate plane with the x axis labeled time in days and the y axis labeled number of boxes, with a line that passes through the points (0, 27), (1,25), (3,21), (4,19) and (7,13)
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Answer: Its D.
Step-by-step explanation:
what happens to the equation of f(x) = 1/x, if it's translated 4 units up and to the right 3 units
Noreen can walk 1/3 mile in 12 minutes. What is her average speed in miles per hour?
Answer:4 mph
Step-by-step explanation:
Greta has two pet lizards, Draco and Lizzy. For every 11 crickets Greta feeds Draco, she always feeds Lizzy 8 crickets. Last week, Greta fed Draco 77 crickets.
Answer:
56 crickets
Step-by-step explanation:
the prices of houses in the us is strongly skewed to the right with a mean of $383,500 and a standard deviation of $289,321. a real estate agent takes a random sample of 30 houses and records the mean price. what is the best description for the sampling distribution? skewed to the right with a mean of 383,500 and a standard deviation of 52,823 skewed to the right with a mean of 383,500 and a standard deviation of 289,321 approximately normal with a mean of 383,500 and a standard deviation of 52,823 approximately normal with a mean of 383,500 and a standard deviation of 289,321
The sampling distribution of the mean price of 30 houses is a normal distribution with a mean of $383,500 and a standard deviation of $52,823 is the correct description among the following given for sampling distribution.
A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
This can be calculated as:
Standard Deviation of Sampling Distribution =
(Standard Deviation of Population) / (√Number of Samples)
= $289,321 / √30
= $289,321 / 5.4772
= $52.823
Therefore, the sampling distribution of the mean price of 30 houses is a normal distribution with a mean of $383,500 and a standard deviation of $52,823.
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