The point on the plane 4x + 7y + 8z - 14 = 0 closest to the point (-2,2,3) is (-1, 1, 1). This can be answered by the concept of Distance formula.
To find the point on the plane closest to the given point, we can use the formula for the distance between a point and a plane. The formula states that the distance between a point (x1, y1, z1) and a plane Ax + By + Cz + D = 0 is given by:
Distance = |Ax1 + By1 + Cz1 + D| / √(A² + B² + C²)
In this case, the coefficients of x, y, and z in the equation of the plane are A = 4, B = 7, and C = 8, and D = -14. The given point is (-2, 2, 3), so x1 = -2, y1 = 2, and z1 = 3.
Plugging these values into the formula, we get:
Distance = |4(-2) + 7(2) + 8(3) - 14| / √(4² + 7² + 8²)
Simplifying, we get:
Distance = 35 / √(189)
To find the point on the plane closest to the given point, we need to minimize this distance. Since the distance is a positive value, the point on the plane closest to the given point will be the one that minimizes the numerator of the distance formula, which is |4(-2) + 7(2) + 8(3) - 14|. This occurs when the numerator is equal to zero.
Setting the numerator equal to zero, we get:
4(-2) + 7(2) + 8(3) - 14 = 0
Solving for x, y, and z, we get:
x = -1
y = 1
z = 1
Therefore, the point on the plane closest to the given point (-2, 2, 3) is (-1, 1, 1).
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What is the factored form of the polynomial?
x2 − 12x + 27?
(x + 4)(x + 3)
(x − 4)(x + 3)
(x + 9)(x + 3)
(x − 9)(x − 3)
Answer:
-9?
Step-by-step explanation:
[infinity]consider the series ∑ 1/n(n+2)n=1 determine whether the series converges, and if it converges, determine its value.Converges (y/n) = ___Value if convergent (blank otherwise = ____
The value of the series is: ∑ 1/n(n+2) = lim N→∞ S(N) = 1/2.
The series ∑ 1/n(n+2)n=1 converges. To determine its value, we can use the partial fraction decomposition:
1/n(n+2) = 1/2 * (1/n - 1/(n+2))
Using this decomposition, we can rewrite the series as:
∑ 1/n(n+2) = 1/2 * (∑ 1/n - ∑ 1/(n+2))
The first series ∑ 1/n is the harmonic series, which diverges. However, the second series ∑ 1/(n+2) is a shifted version of the harmonic series, and it also diverges. But since we are subtracting a divergent series from another divergent series, we can use the limit comparison test to determine whether the original series converges or diverges. Specifically, we can compare it to the series ∑ 1/n, which we know diverges. This gives:
lim n→∞ 1/n(n+2) / 1/n = lim n→∞ (n+2)/n^2 = 0
Since the limit is less than 1, we can conclude that the series ∑ 1/n(n+2) converges. To find its value, we can evaluate the partial sums:
S(N) = 1/2 * (∑_{n=1}^N 1/n - ∑_{n=1}^N 1/(n+2))
= 1/2 * (1/1 - 1/3 + 1/2 - 1/4 + ... + 1/(N-1) - 1/(N+1))
As N approaches infinity, the terms in the parentheses cancel out except for the first and last terms:
S(N) → 1/2 * (1 - 1/(N+1))
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find the partial derivatives of the function f(x,y)=xye−9y
The partial derivatives of the function f(x,y) = xy*e^(-9y) with respect to x and y are: ∂f/∂x = ye^(-9y), and ∂f/∂y = x(-9y*e^(-9y)) + e^(-9y).
The first partial derivative concerning x is obtained by treating y as a constant and differentiating concerning x. The result is ye^(-9y), which means that the rate of change of f concerning x is equal to ye^(-9y).
The second partial derivative concerning y is obtained by treating x as a constant and differentiating concerning y. The result is x(-9ye^(-9y)) + e^(-9y), which means that the rate of change of f concerning y is equal to x times -9ye^(-9y) plus e^(-9y).
To better understand these partial derivatives, we can analyze the behavior of the function f(x,y) = xy*e^(-9y). As we can see, the function is the product of three terms: x, y, and e^(-9y). The term e^(-9y) represents a decreasing exponential function that approaches zero as y increases. Therefore, the value of f(x,y) decreases as y increases. The terms x and y represent a linear function that increases as x and y increase. Therefore, the value of f(x,y) increases as x and y increase.
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A factory makes boxes of cereal. Each box contains cereal pieces shaped like hearts, stars,
and rings.
An employee at the factory wants to check the quality of a sample of cereal pieces from a box.
Which sample is most representative of the population?
Answer:
Step-by-step explanation:
d
The most representative sample of the population would be a random sample of cereal pieces from the box. Therefore, option D is the correct answer.
What is random sampling?In statistics, a simple random sample is a subset of individuals chosen from a larger set in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample in a random way.
Given that, a factory makes boxes of cereal. Each box contains cereal pieces shaped like hearts, stars and rings.
The most representative sample of the population would be a random sample of cereal pieces from the box. This means that the employee should select pieces from the box without looking at or attempting to select any particular shape. This ensures that the sample accurately reflects the distribution of cereal pieces in the box, and gives an accurate representation of the population.
Therefore, option D is the correct answer.
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Simplify the radical expression. Show all your steps.
√363 − 3√27
Answer: simplified expression is 2√3.
Step-by-step explanation:
√363 = √(121 × 3) = √121 × √3 = 11√3
√27 = √(9 × 3) = √9 × √3 = 3√3
√363 − 3√27 = 11√3 − 3(3√3) = 11√3 − 9√3 = 2√3
The simplified form of the given radical expression is 2√3.
What is radical form?Radical form is the expression that involves radical signs such as square root, cube root, etc instead of using exponents to describe the same entity.
The given expression is √363 − 3√27.
Here, √121×3 − 3√9×3
= 11√3-9√3
= 2√3
Therefore, the simplified form of the given radical expression is 2√3.
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For two programs at a university, the type
of student for two majors is as follows.
Find the probability a student is a science major,
given they are a graduate student.
Answer:
Step-by-step explanation:
To find the probability that a student is a science major given that they are a graduate student, we need to use Bayes' theorem:
P(Science | Graduate) = P(Graduate | Science) * P(Science) / P(Graduate)
We know that P(Science) = 0.45 and P(Liberal Arts) = 0.55, and that P(Graduate | Science) = 0.35 and P(Graduate | Liberal Arts) = 0.25. We also know that the total probability of being a graduate student is:
P(Graduate) = P(Graduate | Science) * P(Science) + P(Graduate | Liberal Arts) * P(Liberal Arts)
Plugging in the values, we get:
P(Graduate) = 0.35 * 0.45 + 0.25 * 0.55 = 0.305
Now we can calculate the probability of being a science major given that the student is a graduate student:
P(Science | Graduate) = 0.35 * 0.45 / 0.305 = 0.515
Therefore, the probability that a student is a science major, given they are a graduate student, is approximately 0.515.
Answer:
0.72
Step-by-step explanation:
trust me
suppose x is a continuous variable with the following probability density: f(x)={c(10−x)2, if 0
Probability density function for the continuous variable x is:
f(x) = (3/1000)(10-x)², if 0
Total area under the probability density function is equal to 1.
So, we integrate the function from 0 to 10:
∫[0,10] c(10−x)2 dx
= c ∫[0,10] (10−x)2 dx
= c [-(10-x)³/³] evaluated from 0 to 10
= c [(0-(-1000/3))]
= c (1000/3)
Since the area under the probability density function is equal to 1, we have:
∫[0,10] c(10−x)2 dx = 1
Puting the value of the integral:
c (1000/3) = 1
Solving for c, we get:
c = 3/1000
Therefore, the probability density function for the continuous variable x is:
f(x) = (3/1000)(10-x)², if 0
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Around the beginning of the 1800’s, the population of the U.S. was growing at a rate of about 1.33^t million people per decade, with "t" being measured in decades from 1810.
If the population P(t) was 7.4 million people in 1810, estimate the population in 1820 (one decade later) by considering the work in example 2.
We can determine the population in 1820 was 8.5753 using a linear equation.
What does a linear equation mean in mathematics?A linear equation is one that has just a constant and a first order (linear) component, like y=mx+b, where m is the slope and b is the y-intercept.
When x and y are the variables, the aforementioned is sometimes referred to as a "linear equation of two variables."
dp/dt = [tex]1.37^{t}[/tex]
Integrate both sides.
p[h] = ( [tex]1.37^{t}[/tex])/In (1.37) + c
1810 ⇒ t = 0
7.4 = 1/In (1.37) + C
C = 4.2235
p(H) = ( [tex]1.37^{t}[/tex])/In (1.37) + 4.2235
P (1) = [tex]1.37^{t}[/tex]In (1.37) + 4.2235
= 8.5753
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Please help if you can, i don't understand
Answer: I believe -2 is the answer
Step-by-step explanation: To solve for the function over an interval, you need to know the equation of the function. If you have the equation, you can plug in the values of the interval into the equation to find the corresponding y-values. For example, if the function is y = 2x + 1 and the interval is [0,3], you can plug in x = 0 and x = 3 to find the corresponding y-values and get the ordered pairs (0,1) and (3,7).
Evaluate the expression 7 + 2 x 8 − 5. (1 point)
18
20
48
63
Quickly answer please!
The graph of a function contains the points (-5, 1), (0,
3), (5, 5). Is the function linear? Explain.
(Photo of answer choice included)
(d) The function (-5, 1), (0, 3), (5, 5) is not a linear function
Calculating the type of the functionFrom the question, we have the following parameters that can be used in our computation:
(-5, 1), (0, 3), (5, 5).
A linear function has a constant rate of change, meaning that the slope of the line is always the same.
However, if we plot the given points on a graph, we can see that they do not lie on a straight line.
Therefore, the function is not linear.
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what is the list after the second outer loop iteration?[6,9,8,1,7],[],,,
After the second outer loop iteration, the list is [6,1,7,8,9].
To determine the list after the second outer loop iteration, let's assume we're working with a simple bubble sort algorithm. Here are the steps:
1. First outer loop iteration:
- Compare 6 and 9; no swap.
- Compare 9 and 8; swap to get [6,8,9,1,7].
- Compare 9 and 1; swap to get [6,8,1,9,7].
- Compare 9 and 7; swap to get [6,8,1,7,9].
2. Second outer loop iteration:
- Compare 6 and 8; no swap.
- Compare 8 and 1; swap to get [6,1,8,7,9].
- Compare 8 and 7; swap to get [6,1,7,8,9].
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Find an equation of the tangent line to the curve y=8x at the point (2,64)
Equation of the tangent line to the curve y=8x is y = 8x + 48.
How do we need to find the slope of the tangent at that point?Derivative of the curve, we get:
dy/dx = 8
This means that the slope of the tangent line to the curve at any point is 8.
So, at the point (2,64), the slope of the tangent line is 8.
By point-slope form of a line, we will find the equation of the tangent line:
y - y1 = m(x - x1)
where m is the slope and (x1,y1) is the given point.
Plugging in the values, we get:
y - 64 = 8(x - 2)
Simplifying, we get:
y = 8x + 48
Equation of the tangent line to the curve y=8x at the point (2,64) is y = 8x + 48.
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prove that x2 2: x for all x e z.
We have demonstrated that x² ≥ x for all integers x. Therefore, the statement x² ≥ x for all x ∈ Z is true.
What is inequality?An inequality is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made.
To prove that x² ≥ x for all x ∈ Z, we need to show that the inequality holds true for any arbitrary integer value of x.
We can prove this by considering two cases:
Case 1: x ≥ 0
If x ≥ 0, then x² ≥ 0 and x ≥ 0. Therefore, x² ≥ x.
Case 2: x < 0
If x < 0, then x² ≥ 0 and x < 0. Therefore, x² > x.
In either case, we have shown that x² ≥ x for all integers x. Therefore, the statement x² ≥ x for all x ∈ Z is true.
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I think I understand how to do this but the answer I think it is goes past the graph?
The other root of the quadratic equation include the following (-4, 0).
What is the vertex form of a quadratic equation?In Mathematics and Geometry, the vertex form of a quadratic equation is given by this formula:
y = a(x - h)² + k
Where:
h and k represents the vertex of the graph.a represents the leading coefficient.For the given quadratic function, we have;
y = a(x - h)² + k
0 = a(8 - 2)² - 5
0 = 36a - 5
5 = 36a
a = 5/36
Therefore, the required quadratic function in vertex form is given by;
y = 5/36(x - 2)² - 5
0 = 5/36(x - 2)² - 5
5 = 5/36(x - 2)²
36 = (x - 2)²
±6 = x - 2
x = -6 + 2
x = -4.
Other root = (-4, 0).
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Resuelve con proceso:
Un comerciante vende polos, 200 polos a 8 por 2 soles y 300 polos a 5 por 3 soles. ¿Cual es la diferencia de lo que recibió de la primera venta con la segunda?.
The number of more sole received by merchant in the second sale compared to first sale is equal to 130 soles
Let us first calculate the cost of one pole in each sale.
For the first sale, 8 poles cost 2 soles. So, one pole costs.
2 soles / 8 poles = 0.25 soles/pole
For the second sale, 5 poles cost 3 soles. So, one pole costs.
3 soles / 5 poles = 0.6 soles/pole
Next, let us find out the total revenue from each sale.
For the first sale,
The merchant sold 200 poles. If one pole costs 0.25 soles, then 200 poles would cost.
200 poles × 0.25 soles/pole = 50 soles
For the second sale,
The merchant sold 300 poles. If one pole costs 0.6 soles, then 300 poles would cost.
300 poles × 0.6 soles/pole = 180 soles
The difference between what the merchant received from the first sale and the second sale is,
180 soles - 50 soles = 130 soles
Therefore,, the merchant received 130 soles more from the second sale than the first sale.
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let f (x) = 5x and g(x) = x^1/3. find (fg) (x)
(fg)(x) =
The value of the function (fg)(x) = = ∛5
What is a function?A function can be described as an equation or expression that is used to show the relationship between two variables.
The two variables are;
The dependent variableThe independent variableFrom the information given, we have that;
f(x) = 5x
g(x) = x^1/3
To determine the composite function (fg)(x), substitute the value of(x) as the value of x in the function g(x), we have;
(fg)(x) = 5^1/3
This is written as;
(fg)(x) =(∛5)¹
(fg)(x) = = ∛5
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hannah invested $500 into an account with a 6.5% intrest rate compounded monthly. how much will hannahs investment be worth in 10 years.
Answer:
Using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal (initial amount of investment)
r = the interest rate (as a decimal)
n = the number of times per year the interest is compounded
t = the time (in years)
Plugging in the values:
P = $500
r = 6.5% = 0.065
n = 12 (compounded monthly)
t = 10
A = 500(1 + 0.065/12)^(12*10)
A = $935.98
Hannah's investment will be worth $935.98 after 10 years.
Please solve this geometry problem.
hope this helps you .
(b) region r is the basRegion R is the base of a soli., each cross section perpendicular to the x axis is a semi circle. Write, but do not evaluate, an integral expression that would compute the volume of the solid
of a
An integral expression that would compute the volume of the solid is [tex]V = \int\limits^a_b {1/2 \pi [R(x)]^2} \, dx[/tex]
What is integral expression?An integral expression is a mathematical statement that represents the area under a curve or the volume of a solid in three-dimensional space. It is written using integral notation, which involves an integral sign, a function to be integrated, and limits of integration.
According to given information:If each cross section perpendicular to the x-axis is a semicircle, then the radius of each cross section depends on the x-coordinate of the center of the cross section. Let R(x) be the radius of the cross section at x.
To find the volume of the solid, we can integrate the area of the cross section over the interval of x that defines the base R. The area of each cross section is given by the formula for the area of a semicircle:
[tex]A(x) = (1/2)[/tex][tex]\pi[R(x)]^2[/tex]
The volume of the solid can be found by integrating A(x) over the base R:
[tex]V = \int\limits^a_b {1/2 \pi [R(x)]^2} \, dx[/tex]
where a and b are the limits of integration for x that define the base R.
Note that we are integrating with respect to x, so we need to express the radius R(x) in terms of x.
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given f(7)=2, f′(7)=11, g(7)=−1, and g′(7)=9, find the values of the following. (a) (fg)′(7)= number (b) (fg)′(7)= number
Answer:
Step-by-step explanation:
Solve the equation x² + 4x - 11 = 0 by completing the square.
Fill in the values of a and b to complete the solutions.
x = a - (squared)b
x = a + (squared) b
The required values are -2+√15, -2-√15.
What is a quadratic equation?
Any equation in algebra that can be written in the standard form where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation.
Here, we have
Given: x² + 4x - 11 = 0
we have to find the values of a and b to complete the solutions.
The given equation is x² + 4x - 11 = 0
The general form of a quadratic equation is ax² + bx + c = 0
Comparing with the given equation we have
a = 1
b = 4
c = -11
Rearranging the equation:
x² + 4x = 11
Finding (b/2)²
(4/2)² = 4
Adding to both sides of the equation
x² + 4x + 4 = 11 + 4
(x+2)² = 15
x + 2 = ±√15
x = -2 ±√15
Hence, the required values are -2+√15, -2-√15.
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in hypothesis testing, making decision that that causes a false alarm is equivalent to a. correct decision b. null hypothesis c. type-1 error d. type-2 error
In hypothesis testing, making a decision that causes a false alarm is equivalent to committing a type-1 error.
In hypothesis testing, making a decision that causes a false alarm is equivalent to a Type-1 error. This occurs when we reject the null hypothesis even though it is actually true. It is important to control the probability of making type-1 errors, as this can lead to incorrect conclusions and wasted resources. The correct decision in hypothesis testing is to either accept or reject the null hypothesis based on the evidence presented. A type-2 error, on the other hand, occurs when we fail to reject the null hypothesis even though it is false.
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Help AGAIN!
Which one cheaper and by how much?
View attachment below
Answer: Website A is cheaper, by an amount of, £0.29.
Step-by-step explanation: Here, the problem is simply about, initially adding, and then finding difference between the added results.
That is,
For Website A,
Net Cost = £49.95 + £4.39
= £54.34
Similarly,
For Website B,
Net Cost = £47.68 + £6.95
= £54.63
Therefore, we can clearly see,
Website A is cheaper by,
£(54.63 - 54.34) = £0.29
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3.48 Referring to Exercise 3.39, find
(a) f(y|2) for all values of y;
(b) P(Y = 0 | X = 2).
this is 3.39
3.39 From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample, find (a) the joint probability distribution of X and Y ; (b) P[(X, Y ) ∈ A], where A is the region that is given by {(x, y) | x + y ≤ 2}.
Referring to Exercise 3.39,
(a) f(y|2) for all values of y is f(2|2) = P(Y=2|X=2) = P(X=2, Y=2) / P(X=2) = (1/14) / (3/14) = 1/3
(b) P(Y = 0 | X = 2) = 1
To find f(y|2), we need to first calculate the conditional probability of Y=y given that X=2, which we can do using the joint probability distribution we found in part (a) of Exercise 3.39:
P(Y=y|X=2) = P(X=2, Y=y) / P(X=2)
We know that P(X=2) is equal to the probability of selecting 2 oranges out of 4 fruits, which can be calculated using the hypergeometric distribution:
P(X=2) = (3 choose 2) * (2 choose 0) / (8 choose 4) = 3/14
To find P(X=2, Y=y), we need to consider all the possible combinations of selecting 2 oranges and y apples out of 4 fruits:
P(X=2, Y=0) = (3 choose 2) * (2 choose 0) / (8 choose 4) = 3/14
P(X=2, Y=1) = (3 choose 2) * (2 choose 1) / (8 choose 4) = 3/14
P(X=2, Y=2) = (3 choose 2) * (2 choose 2) / (8 choose 4) = 1/14
Therefore, f(y|2) is:
f(0|2) = P(Y=0|X=2) = P(X=2, Y=0) / P(X=2) = (3/14) / (3/14) = 1
f(1|2) = P(Y=1|X=2) = P(X=2, Y=1) / P(X=2) = (3/14) / (3/14) = 1
f(2|2) = P(Y=2|X=2) = P(X=2, Y=2) / P(X=2) = (1/14) / (3/14) = 1/3
To find P(Y=0|X=2), we can use the conditional probability formula again:
P(Y=0|X=2) = P(X=2, Y=0) / P(X=2) = 3/14 / 3/14 = 1
Therefore, P(Y=0|X=2) = 1.
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2. find the angle in the figure in both radion measure and
angle measure.
ест
6
5cm
The measure of the central angle is 86 degrees.
How to find the central angle?The length of the arc is 9 cm and the radius is 6 centimetres. Therefore, let's find the central angle as follows:
Hence,
length of an arc = ∅ / 360 × 2πr
where
r = radius∅ = central angleTherefore,
length of arc = 9 cm
radius = 6 cm
Therefore,
9 = ∅ / 360 × 2 × 3.14 × 6
9 = 37.68∅ / 360
cross multiply
3240 = 37.68∅
divide both sides by 37.68
∅ = 3240 / 37.68
∅ = 85.9872611465
∅ = 86 degrees.
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Help please!
5/8 ÷ 1/8
Answer: 5
5/8/1/8, you can do 5x8 and also do 8x1 because you can not divide fractions after that you get 40/8 then you divide 40/8 is 5 so the answer is 5
find the limit of the following sequence or determine that the sequence diverges. {tan^−1( 4n/ 4n +5)}
The limit of the given sequence is π/4, and the sequence converges to this value.
The given sequence is {tan^−1(4n/(4n+5))}. To determine if the sequence converges or diverges, we can analyze the limit of the function as n approaches infinity.
As n goes to infinity, the function behaves like tan^−1(4n/4n), which simplifies to tan^−1(1). Since the arctangent function has a range of (-π/2, π/2), tan^−1(1) falls within this range, and it is equal to π/4 (or 45° in degrees).
Now, let's consider the difference between the given function and the simplified one: (4n+5) - 4n = 5. As n becomes larger, the effect of the constant term 5 becomes negligible. Consequently, the function approaches tan^−1(1) as n approaches infinity.
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1. Solve the problem. If the price charged for a bolt is p cents, then x thousand bolts will be sold in a certain hardware store, where p=63-x/20. How many bolts must be sold to maximize revenue A) 630 thousand bolts B) 630 bolts C) 1260 bolts D) 1260 thousand bolts
A total of 630 thousand bolts must be sold to maximize revenue. The correct answer is A) 630 thousand bolts.
To maximize revenue, we need to first determine the revenue function.
Revenue is given by the product of price (p) and quantity (x).
In this case, p = 63 - x/20.
Write the revenue function:
R(x) = px
= (63 - x/20)x
Simplify the function:
R(x) = 63x - (x²)/20
To maximize the revenue, find the vertex of the parabola formed by the quadratic function.
The x-coordinate of the vertex is given by -b/(2a), where a and b are the coefficients of x² and x, respectively.
In this case, a = -1/20 and b = 63. So, the x-coordinate of the vertex is:
x = -63 / (2 (-1/20))
= 63 (20 / 2)
= 630.
Therefore, option A) is correct.
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Given: A_n = 30/3^n Determine: (a) whether sigma _n = 1^infinity (A_n) is convergent. _____
(b) whether {An} is convergent. _____
If convergent, enter the limit of convergence. If not, enter DIV.
As n increases, 3^n becomes larger, making the fraction 30/3^n approach zero. Therefore, the sequence {A_n} is convergent, and the limit of convergence is 0. (a) Σ(A_n) is convergent and (b) {A_n} is convergent with the limit of convergence equal to 0.
(a) To determine whether sigma _n = 1^infinity (A_n) is convergent, we need to take the sum of the sequence A_n from n=1 to infinity:
sigma _n = 1^infinity (A_n) = A_1 + A_2 + A_3 + ...
Substituting A_n = 30/3^n, we get:
sigma _n = 1^infinity (A_n) = 30/3^1 + 30/3^2 + 30/3^3 + ...
To simplify this, we can factor out a common factor of 30/3 from each term:
sigma _n = 1^infinity (A_n) = 30/3 * (1/3^0 + 1/3^1 + 1/3^2 + ...)
Now, we recognize that the expression in parentheses is a geometric series with first term a=1 and common ratio r=1/3. The sum of an infinite geometric series with first term a and common ratio r is:
sum = a / (1 - r)
Applying this formula to our series, we get:
sigma _n = 1^infinity (A_n) = 30/3 * (1/ (1 - 1/3)) = 30/2 = 15
Therefore, sigma _n = 1^infinity (A_n) is convergent, with a limit of 15.
(b) To determine whether {An} is convergent, we need to take the limit of the sequence A_n as n approaches infinity:
lim n->infinity (A_n) = lim n->infinity (30/3^n) = 0
Therefore, {An} is convergent, with a limit of 0.
(a) To determine if the series Σ(A_n) from n=1 to infinity is convergent, we can use the ratio test. The ratio test states that if the limit as n approaches infinity of the absolute value of the ratio A_(n+1)/A_n is less than 1, the series converges.
For A_n = 30/3^n, we have:
A_(n+1) = 30/3^(n+1)
Now let's find the limit as n approaches infinity of |A_(n+1)/A_n|:
lim(n→∞) |(30/3^(n+1))/(30/3^n)| = lim(n→∞) |(3^n)/(3^(n+1))| = lim(n→∞) |1/3|
Since the limit is 1/3, which is less than 1, the series Σ(A_n) converges.
(b) To determine if the sequence {A_n} is convergent, we need to find the limit as n approaches infinity:
lim(n→∞) (30/3^n)
As n increases, 3^n becomes larger, making the fraction 30/3^n approach zero. Therefore, the sequence {A_n} is convergent, and the limit of convergence is 0.
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