To find the area inside the larger loop and outside the smaller loop of the limaçon r = 1 2 + cos(θ), we need to first visualize the graph of the limaçon.
The equation r = 1 2 + cos(θ) represents a curve that resembles a snail shell or a heart shape with a loop inside another loop. To find the area inside the larger loop and outside the smaller loop, we need to set up the integral using the polar coordinates.
The formula for the area enclosed by a polar curve is given by:
A = 1/2 ∫(θ2-θ1) (r2-r1)² dθ, where r2 is the outer curve, r1 is the inner curve, and θ1 and θ2 are the angles where the curves intersect.
In this case, the inner curve is r = 1 - cos(θ), and the outer curve is r = 1/2 + cos(θ). The curves intersect when cos(θ) = 1/2 or θ = π/3 and θ = 5π/3.
So, we need to split the integral into two parts, one for θ = π/3 to θ = 5π/3, and another for θ = 5π/3 to θ = π/3.
This is because the outer curve becomes the inner curve and vice versa when we cross the angle θ = 5π/3. For the area inside the larger loop and outside the smaller loop, we need to subtract the area enclosed by the inner curve from the area enclosed by the outer curve.
Using the formula above and plugging in the values, we get:
A = 1/2 ∫(5π/3-π/3) [(1/2 + cos(θ))² - (1-cos(θ))²] dθ
Simplifying this integral, we get:
A = 1/2 ∫(5π/3-π/3) [5/4 + 2cos(θ)] dθ
A = 5/8 [θ + sin(θ)](5π/3-π/3)
A = 5/8 [4π/3 + sin(4π/3) - (π/3 + sin(π/3))]
A = 5/8 [4π/3 - √3/2]
A = 5π/6 - (5/16)√3
Therefore, the area inside the larger loop and outside the smaller loop of the limaçon r = 1 2 + cos(θ) is 5π/6 - (5/16)√3.
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To find the area inside the larger loop and outside the smaller loop of the limaçon r = 1 + 2cos(θ), use the formula for finding the area enclosed by a polar curve.
Explanation:122π/3∫0(12+cosθ)2dθ, r=(1/2)+cos\u0026#952, , 1, 2, cos, , FindThe general formula is A = (1/2)∫(r^2)dθ, where r is the equation of the curve. In this case, we need to find the area between two curves: the larger loop given by r = 1 + 2cos(θ) and the smaller loop given by r = 1. To find the limits of integration, we need to find the values of θ where the two curves intersect. After finding the values of θ where the curves intersect, we can integrate the difference between the two equations r = 1 + 2cos(θ) and r = 1 with respect to θ.
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Graphing quadratic equations
A graph of the given quadratic equation is shown below.
The axis of symmetry is equal to 3.
The vertex is equal to (3, -1).
What is the graph of a quadratic function?In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the graph of this quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).
Since the leading coefficient (value of a) in the given quadratic function y = x² - 6x + 8 is positive 1, we can logically deduce that the parabola would open upward and the solution would be on the x-intercepts. Also, the value of the quadratic function f(x) would be minimum at -1.
In conclusion, the axis of symmetry is 3 while the vertex is given by the ordered pair (3, -1).
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Anybody know how to do this?
The point which is a solution to the system y = x + 7 and y = 11 - x is: 2. (2, 9).
The value of b in the second equation must be: 1. 10.
The x-coordinate of the solution to the system is: 4. x = -2.
The y-coordinate of the solution is: 4. y = 4.
The point (2, 5) would not be a solution to this equation: 4. 2x + 4y = 12
How to determine the solution to the system of equations?In this scenario and exercise, you are required to determine the solution to the given system of equations as follows;
y = x + 7
y = 11 - x
By using the substitution method, we have the following:
x + 7 = 11 - x
2x = 4
x = 4/2 = 2.
y = x + 7 = 2 + 7 = 9.
Question 2:Next, we would determine the value of b as follows;
x = 1
y = 3x + 5 = 3(1) + 5 = 8
y = -2x + b
b = y + 2x = 8 + 2(1) = 10.
Question 3:For the x-coordinate of the solution to the system, we have:
y = 4x + 7
x + y = -3
-3 - x = 4x + 7
5x = -10
x = -10/5 = -2.
Question 4:For the y-coordinate of the solution, we have:
x + 3 = -2x + 6
3x = 3
x = 1
y = x + 3 = 1 + 3 = 4
Question 5:2x + 4y = 12
2(2) + 4(5) = 12
4 + 20 = 12
24 = 12 (False).
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A first number plus twice a second number is 8. . Twice the first number plus the second totals 34
Answer:
20-6Step-by-step explanation:
You want two numbers such that ...
first plus twice the second is 8twice the first plus the second is 34.SetupLet x and y represent the first and second numbers, respectively. Then the given relations can be written as the equations ...
x +2y = 8
2x +y = 34
SolutionSolving the first equation for x gives ...
x = 8 -2y
Using that in the second equation, we have ...
2(8 -2y) +y = 34
16 -3y = 34 . . . . . . eliminate parentheses
-18 = 3y . . . . . . . . add 3y-34
-6 = y . . . . . . . . divide by 3
x = 8 -2(-6) = 8 +12 = 20
The first number is 20, and the second number is -6.
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Suppose someone claims the average delivery time for u.s. packages to be delivered during december is 2 days. you believe it takes longer than that. you conduct a hypothesis test; what are your hypotheses?
The null hypothesis (H0) is that the average delivery time for U.S. packages during December is 2 days, while the alternative hypothesis (Ha) is that the average delivery time is longer than 2 days.
Based on your question, you want to test if the average delivery time for U.S. packages in December is longer than 2 days. In this hypothesis test, you would have the following hypotheses:
Null Hypothesis (H0): The average delivery time for U.S. packages in December is 2 days.
Alternative Hypothesis (H1): The average delivery time for U.S. packages in December is greater than 2 days.
Thus, H0: μ ≤ 2 (where μ is the population mean delivery time)
Ha: μ > 2
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If we want to decrease the type ii error in a hypothesis test while maintaining type i error at its current level we should:_______
To decrease the type II error in a hypothesis test while maintaining the type I error at its current level, we need to increase the sample size.
What is detailed explaination of the given answer?Type I error shows when we reject a null hypothesis which is actually true. Type II error shows when we fail to reject a null hypothesis which is actually false.
Increasing the sample size can decrease the probability of making a Type II error, as it increases the power of the hypothesis test. Probability of correctly rejecting a false null hypothesis is called power.
By increasing the sample size, we can increase the power of the test while maintaining the significance level (and thus the type I error) at its current level.
This means that we will be less likely to miss a true effect (i.e., lower the probability of making a Type II error) while still controlling the probability of false positives (i.e., Type I error) at the desired level.
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in the standard normal distribution, what is the proportion of values between negative infinity and 1 z-score?
Thus, the proportion of values between negative infinity and 1 z-score in the standard normal distribution is about 84.13%.
In the standard normal distribution, the proportion of values between negative infinity and 1 z-score can be determined using the cumulative distribution function (CDF) of the normal distribution.
The standard normal distribution, also known as the Z-distribution, has a mean (µ) of 0 and a standard deviation (σ) of 1. The proportion of values between negative infinity and any z-score corresponds to the area under the curve of the standard normal distribution up to that z-score.To find the proportion of values between negative infinity and 1 z-score, you can consult a Z-table or use a statistical calculator or software that provides the CDF for the standard normal distribution. The CDF value for a z-score of 1 is approximately 0.8413.Therefore, the proportion of values between negative infinity and 1 z-score in the standard normal distribution is about 84.13%. This means that 84.13% of the data points lie to the left of a z-score of 1 in a standard normal distribution.Know more about the cumulative distribution function (CDF)
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In the diagram below, the dimensions of the large rectangle are (3x-1) by (3x+7) units. The Demi sound of the cut-out rectangle are x by 2x+5 units. Write a simplified expression for the area of the shaded region, in square units.
The expression that represents the area of shaded region is expressed as:
How to Write the Expression for the Area of the Shaded Region?The area of the shaded region = area of larger rectangle - area of smaller rectangle.
Area of a rectangle = length * width
Area of shaded region = [(3x + 7)(3x - 1)] - [(2x + 5)(x)]
Area of shaded region = (9x² + 18x - 7) - (2x² + 5x)
Open the bracket:
9x² + 18x - 7 - 2x² - 5x
Combine like terms:
9x² - 2x² + 18x - 5x - 7
Area of shaded region = 7x² + 13x - 7
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If c is a nonzero constant such that x^2+cx+9c is equal to the square of a binomial, then what is c?
Answer:
c = 6.
Step-by-step explanation:
x^2+cx+9 = (x + b)^2 where b is a constant
x^2 + cx + 9 = x^1 + 2bx + b^2
so 2b = c
and b^2 = 9
So b = 3
and c = 2*b = 6
To solve this problem, we will use the fact that the square of a binomial (a+b)^2 is equal to a^2 + 2ab + b^2.
Let's start by assuming that x^2 + cx + 9c is the square of a binomial (a+b)^2. Expanding (a+b)^2 using the formula above, we get:
(a+b)^2 = a^2 + 2ab + b^2
If we compare this to x^2 + cx + 9c, we see that:
a^2 = x^2 (1)
2ab = cx (2)
b^2 = 9c (3)
From (1), we get a = ±√x^2 = ±x.
Substituting a = ±x into (2), we get:
2(±x)b = cx
Simplifying this equation, we get:
b = ±(c/2)
Substituting a = ±x and b = ±(c/2) into (3), we get:
x^2 + (c/2)^2 = 9c
Multiplying both sides by 4, we get:
4x^2 + c^2 = 36c
Rearranging this equation, we get:
4x^2 - 36c + c^2 = 0
This is a quadratic equation in c. Using the quadratic formula, we get:
c = (36 ± √(36^2 - 4(4)(c^2))) / (2(4))
Simplifying this equation, we get:
c = (9 ± √(81 - c^2)) / 2
Since c is nonzero, we can ignore the solution c = 0. Therefore, we have:
c = (9 + √(81 - c^2)) / 2 or c = (9 - √(81 - c^2)) / 2
We can check that both solutions satisfy the original condition that x^2 + cx + 9c is the square of a binomial.
However, we need to ensure that the square root in each solution is real, which means that 81 - c^2 ≥ 0.
For the first solution, we have:
81 - c^2 ≥ 0
c^2 ≤ 81
|c| ≤ 9
Therefore, the first solution is valid for c such that -9 ≤ c ≤ 9.
For the second solution, we have:
81 - c^2 ≥ 0
c^2 ≤ 81
|c| ≥ 9
However, we already know that c is nonzero, so |c| > 0. Therefore, the second solution is not valid for any value of c.
Therefore, the only valid solution is:
c = (9 + √(81 - c^2)) / 2, for -9 ≤ c ≤ 9.
I hope this helps! Let me know if you have any questions.
If x^2 + cx + 9c is equal to the square of a binomial, then it can be written in the form (x + A)^2, where A is a constant. Expanding the square of the binomial, we get:
(x + A)^2 = x^2 + 2Ax + A^2
Comparing this with the given expression, x^2 + cx + 9c, we can deduce the following:
2A = c and A^2 = 9c
We are asked to find the value of c. Using the first equation (2A = c), we can substitute c in the second equation:
A^2 = 9(2A)
A^2 = 18A
Since A is nonzero, we can divide both sides by A:
A = 18
Now, using the first equation, we can find the value of c:
c = 2A
c = 2(18)
c = 36
Therefore, the value of c is 36.
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14+5x -11= 7x+3-2x
what is it
The equation "14 + 5x -11 = 7x + 3 - 2x" has infinitely many solutions, so any value of "x" will satisfy the equation.
In order to solve for "x" in the equation "14 + 5x - 11 = 7x + 3 - 2x", we need to simplify the expression by combining like terms:
14 + 5x - 11 = 7x + 3 - 2x
Simplifying the LHS,
We get,
3 + 5x = 7x + 3 - 2x
Simplifying the RHS,
We get,
3 + 5x = 5x + 3
Subtracting 5x from both sides,
We get,
3 = 3
The equation simplifies to 3 = 3, which is always true.
This means that the equation has infinitely many solutions and that any value of "x" will satisfy the equation.
Therefore, there is no unique value of "x" that can be found from the equation.
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The given question is incomplete, the complete question is
Find the value of "x" in the expression "14+5x -11= 7x+3-2x".
a measure of meaningfulness that expresses the difference between the experimental and control group in standard deviation units is the
The measure you're referring to is called "Cohen's d." It expresses the difference between the experimental and control group in standard deviation units, providing an indicator of effect size and meaningfulness in a study.
The measure of meaningfulness that expresses the difference between the experimental and control group in standard deviation units is the effect size. Effect size is typically calculated by dividing the difference between the means of the two groups by the standard deviation of the control group. It is a useful tool for comparing the effectiveness of different interventions or treatments, as it allows researchers to assess the magnitude of the difference between the groups beyond just statistical significance.
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a raffle for a charity fund-raiser is being planned. each of 2000 raffle tickets will be sold for $1.00 . the holders of 32 winning tickets will each win a prize. the table shows the prize values and the number of prizes for each value. prize value number of prizes $25 20 $50 10 $300 2 the random variable w represents the value of the prize won for a single ticket minus the cost of the ticket. what is the expected value of w ?
The expected value 'w' for the total raffle tickets sold for $1.00 with other conditions is equal to $0.784.
Number of raffle tickets sold for $1.00 = 2000
Expected value of w is ,
= Expected value of the prize won - cost of a single ticket $1.00
The probability of winning a $25 prize is
= 20/2000
= 0.01.
The expected value of winning this prize is equal to
= (25 - 1) × 0.01
= 0.24
The probability of winning a $50 prize is
= 10/2000
= 0.005.
The expected value of winning this prize is equal to
= (50 - 1) × 0.005
= 0.245
The probability of winning a $300 prize is
= 2/2000
= 0.001.
The expected value of winning this prize is,
= (300 - 1) × 0.001
= 0.299
This implies,
The expected value of w is equal to
= 0.24 + 0.245 + 0.299
= 0.784
Therefore, the expected value of winning a prize minus the cost of the ticket is $0.784.
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what sample size should we select if we wish to develop a 90% confidence interval for the average diameter
The sample size required if we wish to develop a 90% confidence interval for the average diameter is more than 0.1 units.
To decide the test estimate required for a 90% certainty interim for the normal distance across, we have to know the taking after :
1. The level of certainty (which is 90% in this case).
2. The standard deviation of the populace (which we'll expect is known).
3. The margin of error (which is the greatest separation between the test cruel and the genuine populace cruel that we're willing to endure).
Assuming we know the standard deviation of the populace, ready to utilize the equation:
n = (z²* σ²) / E²
where:
n is the test measure
z is the z-score compared to the level of certainty (which is 1.645 for 90% certainty)
σ is the standard deviation of the populace
E is the edge of mistake
We ought to select esteem for E, which speaks to the most extreme separation between the test cruel and the genuine populace cruel(mean) that we're willing to endure.
For case, in the event that we need the interim to have a width of no more than 0.1 units, at that point we would select E = 0.1/2 = 0.05.
So, stopping within the values we know, we get:
n = (1.645² * σ²) / E²
In case we accept that the standard deviation of the populace is 0.5 units (fair as a case), at that point we get:
n = (1.645² * 0.5²) / 0.05²
n = 67.65
Adjusting up to the closest numbers, we would require a test estimate of 68 to create a 90% confidence interim for the normal breadth with an edge of blunder of no more than 0.1 units.
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solve for x: 4(x−3) = 96
4(x−3) = 96
x=27
-12 + 12 + 4x = 96 + 12
Combine like terms: -12 + 12 = 0 0 + 4x = 96 + 12
4x = 96 + 12
Combine like terms: 96 + 12 = 108
4x = 108
Divide each side by '4'.
x = 27
Ixl please help please
The ordering of the topics from broader to narrowest are given as follows:
The influence of popular media on elections.The influence of television programs on elections.The influence of televised debates on elections.What are broad and narrow topics?Broad topics are topics that affect a larger amount of the population than narrower topics affect.
In the context of this problem, we have that the television is a subset of the popular media, hence the popular media is broader than the television.
Televised debates are one program on the television, that is, it is a subset of the television, hence television programs are broader than television debated.
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A student says the histogram is skewed right with an outlier and intervals of 5. Is the student correct? If not explain
The student is not correct because there are intervals of 4 and the histogram is relatively symmetric.
Why is the student wrong ?The histogram is not skewed right, but instead appears to be faintly symmetrical; the frequency of points commence from a count of 0, proceeding to pulsate up to 4 and then progressively meander downward from there to 3.
No apparent outlier can be seen in the provided histogram data; this refers to any data point that disagreeably differs from the remainder of datasets. In this occurrence, no numbers stand out significantly deviantly from the rest.
The intervals used in the histogram are not five units wide, but four units wide instead.
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You have the chance to purchase a car for $6,250.
You must put 10% down and finance the balance at 6% for 36 months. Sales tax is 8.25 %.
You have no trade in.
What are the monthly payments?
The monthly payments after the down payment are $186.58 per month.
What are the monthly payments?We will first calculate amount of the down payment which is:
= 10% * 6,250.
= $625.
So, the financed amount for the purchase will be:
= $6,250 - $625
= $5,625.
We will now calculate the amount of sales tax where tax rate is 8.25%, this give us:
= $6,250 * 8.25%
= $515.63.
The total amount financed including sales tax is:
= $5,625 + $515.63
= $6,140.63.
To calculate the monthly payments, we can use the formula M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1 ]
M = $6,140.63* [ 0.005(1 + 0.005)^36 ] / [ (1 + 0.005)^36 – 1 ]
M = $6,140.63* 0.03042193745
M = $186.58
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plsss helpppp 6 grade math
Graph the given data set and describe what kind of model best describes the data. Then write a function that models
the data.
x y
-2,-1
-1,-3
0,-1
1,2
2,7
The graph of the data set is in the image at the end, and it can be modeled by the quadratic:
y = 2*(x + 1)² - 3
How to find a function that models the data?Here we have the table:
x y
-2, -1
-1, -3
0, -1
1, 2
2, 7
The graph of it can be seen in the image at the end, there we can see that this seems to be a quadratic function
At the graph we also can see that (-1, -3) is te vertex, then if the leading coefficient is a, we can write the quadratic equation as:
y = a*(x + 1)² - 3
And we know that the equation passes throug (0, -1), replacing that we will get.
-1 = a*(0 + 1)² - 3
-1 = a - 3
-1 + 3 = a
2 = a
The function is:
y = 2*(x + 1)² - 3
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9.a. Find the H.C.F. of: [3] x² - y² - 2yz - z² and y² - z² - 2zx - x²
Answer:
We can rewrite the given expressions as:
(3x² - y² - 2yz - z²) and (y² - z² - 2zx - x²)
To find the H.C.F., we can use the Euclidean algorithm. We start by dividing the first expression by the second expression:
(3x² - y² - 2yz - z²) ÷ (y² - z² - 2zx - x²)
Using long division or synthetic division, we get:
(3x² - y² - 2yz - z²) = (3x + y + z)(x - y + z) + 2y(x - z)
Therefore, the remainder is 2y(x - z). We can now divide the second expression by this remainder:
(y² - z² - 2zx - x²) ÷ 2y(x - z)
Using long division or synthetic division, we get:
(y² - z² - 2zx - x²) = -x(x - y + z) + z(x - y + z)
Therefore, the remainder is z(x - y + z).
Since the second remainder is not zero, we need to continue with the algorithm. Now we divide the remainder 2y(x - z) by the remainder z(x - y + z):
2y(x - z) ÷ z(x - y + z)
Using long division or synthetic division, we get:
2y(x - z) = 2y(x - y + z) - 2y²
Therefore, the remainder is -2y². Now we divide the previous remainder z(x - y + z) by this new remainder:
z(x - y + z) ÷ (-2y²)
Using long division or synthetic division, we get:
z(x - y + z) = -1(-2y²) + z²
Therefore, the H.C.F. of the original expressions is the absolute value of the last remainder, which is |-2y²| = 2y².
Therefore, the H.C.F. of (3x² - y² - 2yz - z²) and (y² - z² - 2zx - x²) is 2y².
What would the Volume and Surface area of a pentagonal pyramid be if the Apothem is 3 square root 2 and the height is 3
The volume of the Pentagonal pyramid is 8.25 units² and surface area is 12.2 units²
What is volume of a pyramid?A Pyramid is a three-dimensional shape. A pyramid has a polygonal base and flat triangular faces.
The volume of of a pyramid is expressed as;
V = 1/3 b × h
where b is the area of the base
Area of the base = 1/2 nsa
where s is the side length and n is the number of sides and a is the apothem.
side length = apothem × 2tan(180/n)
= √3/2 × 2tan36
= √3 × tan36
= 1.26m
Therefore,
A = 1/2( 5 × 1.26 × √3/2)
A = 1/2 ( 5.5)
A = 2.75 units²
Volume of the pyramid = 2.75 × 3
= 8.25 units²
Area of a face = 1/2bh
= 1/2 × 1.26 × 3
= 3.78/2
= 1.88 units²
Total surface area = 1.88 × 5
= 9.45 units² + 2.75units²
= 12.2 units²
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What is the value of 2/5+3/7
Answer:
The answer is 29/35
Step-by-step explanation:
2/5+3/7
find the LCM of 5 and 7
[7(2)+5(3)]/35
14+15/35=29/35
A 7
th grade class is taking an end-of-year trip to an amusement park. Each student will pay $48
. The cost covers the $22
admission fee, a ticket to the aquatic show, and a ticket to the firework display. The tickets to the aquatic show and fireworks display are the same price.
What is the price of each ticket?
Answer: $13
Step-by-step explanation:
Let x be the price of each ticket (to the aquatic show and fireworks display).
Then, we know that:
$48 = $22 + 2x
Simplifying this equation, we get:
$48 - $22 = 2x
$26 = 2x
x = $13
Therefore, the price of each ticket is $13.
What is the value of x?
Answer:
x=18.75
Step-by-step explanation.
We have to realize that this is a line. A line has 180 degrees.
Therefore we can make the equation;
(6x+30)+2x=180
8x+30=180
8x=150
x=18.75
please mark as branliest
Answer:
x=18.75
Step-by-step explanation:
We can see that these 2 angles are on a straight line, meaning that these 2 angles must add up to 180°.
We can set up an equation:
180=(6x+30)+2x
combine like terms
180=8x+30
subtract 30 from both sides
150=8x
divide both sides by 8
x=18.75
Hope this helps! :)
5. A picture sold for $500 last year.
This year, the picture is valued at $400.
What is the percent of decrease?
Answer:
Above picture is,
the explanation.
What value of X makes the equation 33x=11 true
Answer:
x = 1/3
Step-by-step explanation:
To solve this equation for x, divide each side by 33.
33x/33 = 11/33
x = 1/3
Answer:
x=3
Step-by-step explanation:
The value of x that makes the equation true is 3.
What is an equation?
An equation is a statement that equates to two expressions.
The given equation is:
Divide both sides by 11
So, the required value of x=3.
A family has a unique pattern in their tile flooring on the patio. An image of one of the tiles is shown.
A quadrilateral with a line segment drawn from the bottom vertex and perpendicular to the top that is 5 centimeters. The right vertical side is labeled 3 centimeters. The portion of the top from the left vertex to the perpendicular segment is 5 centimeters. There is a horizontal segment from the left side that intersects the perpendicular vertical line segment and is labeled 6 centimeters.
What is the area of the tile shown?
53 cm2
45.5 cm2
42.5 cm2
36.5 cm2
Edward is making his special punch for the Starry Night homecoming dance.
There is a proportional relationship between the number of cans of peach juice concentrate in a punch bowl, x, and the corresponding number of bottles of lemon-lime soda, y.
x (cans of peach juice concentrate) y (bottles of lemon-lime soda)
2 6
3 9
4 12
5 15
Write an equation for the relationship between x and y. Simplify any fractions.
y=
The equation of the line is written as y = (2/3)x + (14/3).
The equation of a line is a mathematical representation of the relationship between each point's x and y coordinates along a straight line.
Y = mx + b, where m is the line's slope and b is its y-intercept, is its mathematical formula (the point where the line intersects the y-axis). This equation allows us to rapidly calculate the y-coordinate of a point given its x-coordinate.
The equation of the line can be written as,
y = mx + c
The slope of the line is,
m = ( y₂ - y₁) / (x₂ - x₁)
m = ( 6 - 2 ) / ( 9-3 )
m = 2/3
The y-intercept is calculated as,
y = 2/3x + c
6 = (2/3) x 2 + c
c = 14/3
The equation of the line is,
y = 2/3x + 14/3
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a company receives customer satisfaction ratings on a scale from 0 to 10, inclusive. on the first six surveys that the company received, the average (arithmetic mean) of the ratings was 7.7. what is the least rating the company can receive on the seventh survey and still be able to have an average of at least 8 for the first 10 surveys? (round to the nearest tenth.)
The least amount of ratings the particular company can receive is 3 on the seventh survey and still be eligible to have an average of at least 8 for the first 10 surveys.
Here we have to implement basic algebraic application
Let us consider x as the the least rating the company can get on the seventh survey.
The average of the first six surveys is 7.7, so their total score is 6 × 7.7 = 46.2.
The average of at least 8 for the first ten surveys, their total score should be at least 80.
So for the first ten surveys, their total score should be
80 = (46.2 + x + y)/10
Here
y = score for survey 8 and survey 9.
Evaluating for y
y = (80 - 46.2 - x × 2)/2
= (33.8 - x)/2.
As each survey rating is an integer between 0 and 10 inclusive, we need to find the smallest integer value of x that makes y ≥ 0
Then, (33.8 - x)/2 >= 0.
Multiplying both sides by 2
33.8 - x >= 0
Subtracting 33.8 from both sides
-x >= -33.8
Multiplying both sides by -1
x <= 33.8
Hence, the smallest integer value of x that makes y ≥0 is x = 3.
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2. A trolley car track rises vertically 40 feet over a horizontal distance of 630 feet. What is the angle of
elevation of the track?
Step-by-step explanation:
For a right triangle
tan Φ = opposite leg / adjacent leg
tan Φ = 40 / 630
Φ = arctan (40/630) = 3.6 degrees
Questions from Page 45 in Dale Seymour Publications' Developing Skills in Algebra Book B (adding all of this info in case someone needs to keyword it at some point)
4.] The side of one square is five times that of another square. The difference in the areas of the two squares is 96. Find the sum of the areas of the two squares.
6.] The second side of a triangle is four more than the first side, and the third side is five more than the second. If the product of the first and third sides is added to the product of the second and third sides, the result is 124 more than twice the square of the first side. Find the perimeter of the triangle.
7.] A concrete walk 5 m wide is built around a grass court. The length of the court is 12 m more than its width. The area of the walk is 560 m^2. Find the dimensions of the grass court.
The sum of the areas of the two squares = 104
The perimeter of the triangle is 27.67
How to solve for the area1. We have 5x)² - x² = 96
next wer have to solve for x
25x² - x² = 96
x = 2
x² = 2² = 4
5x)² = 10² = 100
sum of the areas = 4 + 100 = 104
2. b = a + 4
c = b + 5
ac + bc = 2a^2 + 124
substitute b and c from the relationships:
When we simplify we have 18a = 88
a = 44/9
b = a + 4 = 44/9 + 4 = 80/9
c = b + 5 = 80/9 + 5 = 125/9
perimeter = a + b + c:
= 44/9 + 80/9 + 125/9
= 249/9 = 27.67
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