a) The sales of the cooking club will be $80, while the expenditures will be $80 when the number of cakes sold will cover their expenditures (break-even point).
b) The number of cakes that the members of the cooking club will sell at the street fair at this point (break-even point) is 10.
What is the break-even point?The break-even point is the sales units that will equate to the expenditures.
At the break-even point, there is no profit or loss because the total sales revenue equals the total costs (fixed and variable).
The selling price per unit = $8
The fixed cost for a booth at the fair = $20
The variable cost price per unit = $6
Contribution margin per unit = $2 ($8 - $6)
The break-even point in units = Fixed Cost/Contribution margin per unit
= 10 units ($20/$2)
Total sales revenue at the break-even point = $80 ($8 x 10)
Total variable cost = $60 ($6 x 10)
Total fixed and variable costs = $80 ($20 + $60)
Thus, at the break-even point, the cooking club will have sold 10 units making revenue to equal the total costs.
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In a certain shipment, the weights of twelve books average 2. 75 pounds. If one of books is removed, the weights of the remaining books average 2. 70 pounds. What was the weight, in pounds, of the book that was removed?
Answer:
Let's call the weight of the book that was removed "x".
The total weight of the twelve books can be represented as 12 times the average weight of 2.75 pounds:
12(2.75) = 33
After the book is removed, there are only 11 books left, and their total weight can be represented as 11 times the new average weight of 2.70 pounds:
11(2.70) = 29.7
We can set up an equation using these two expressions and the weight of the book that was removed:
33 - x = 29.7
Solving for x, we get:
x = 33 - 29.7
x = 3.3
Therefore, the weight of the book that was removed was 3.3 pounds.
Let X and Y be independent, geometrically distributed random
variables, each with parameter p, p ∈(0, 1). Set N = X + Y.
(a) Find the joint PMF of X, Y, and N.
(b) Find the joint PMF of X and N.
(a) The joint PMF of X, Y, and N can be found by multiplying the two independent PMFs of X and Y. The joint PMF is: P(X=x, Y=y, N=n) = P(X=x) * P(Y=y) = (1-p)^x * p * (1-p)^y * p = (1-p)^(x+y) * p^2.
(b) The joint PMF of X and N can be found by marginalizing over Y. The joint PMF is: P(X=x, N=n) = P(X=x, Y=n-x) = (1-p)^(x+(n-x)) * p^2 = (1-p)^n * p^2.
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Simplify the following expression by combining like terms 3y+8+4y+2
the options are
A- 156
B-91
C-26
D-11
The solution is, that central angle Ф is, 3.49 radians.
What is arc length?Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length.
here, we have,
The arc length formula is s = rФ,
where r is the radius and Ф is the central angle.
We know s and r and need to calculate Ф.
now, we get,
From s = rФ
we get Ф = central angle = s/r
Here, that central angle Ф is,
(31.4 cm) / 9 cm
= 3.49 radians
Hence, The solution is, that central angle Ф is, 3.49 radians.
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Suppose a machine produces metal parts that contain some defective parts with probability 0.05. How many parts should be produced in order that the probability of atleast one part being defective is 21 or more?
(Given that, log1095=1.977 and log102=0.3)
O 11
O 12
O 15
O 14
The probability of at least one part being defective is 21 or more when 14 parts are produced. So, the correct answer is D: 14.
Let X be the number of defective parts among n parts produced. Since each part can either be defective or non-defective, X follows a binomial distribution with parameters n and p, where p = 0.05.
We want to find the smallest value of n such that P(X ≥ 1) ≥ 0.21. We can use the complement rule to rewrite this as P(X < 1) ≤ 0.79.
P(X < 1) = P(X = 0) = (1 - p)^n
= (0.95)^n
We need to find n such that (0.95)^n ≤ 0.79. Taking logarithms of both sides, we get:
n log(0.95) ≤ log(0.79)
n ≥ log(0.79) / log(0.95)
n ≥ 13.65
Since we need n to be an integer, we round up to the nearest integer and get n = 14.
Therefore, the answer is option D: 14.
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Find the equation of the line shown. 10 9 8 6 5 4 3 2 1 1 23 4 5 67 8 9 10
The equation of the line shown is y = 2x.
What is a proportional relationship?In Mathematics, a proportional relationship can be defined as a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:
y = kx
Where:
x and y represents the variables or data points.k represents the constant of proportionality.Next, we would determine the constant of proportionality (k) as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 2/1 = 4/2
Constant of proportionality, k = 2.
Therefore, the required equation is given by:
y = kx
y = 2x
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1. Suppose FR" - Rand GR" - Rare linear functions. That is they each preserve vector additions in their own dimension and ca multiplications. Show that G. F is a near function de, that it respects vec
FR" - Rand GR" - Rare linear functions. That is they each preserve vector additions in their own dimension and ca multiplications. G ∘ F is a linear function.
The question is asking you to show that the composition of two linear functions, G and F, is a linear function itself.
To show that G ◦ F is a linear function, we need to show that it satisfies the two properties of linearity: preservation of vector addition and preservation of scalar multiplication.
Let's first consider preservation of vector addition:
(G ◦ F)(u + v) = G(F(u + v)) [by definition of composition]
= G(F(u) + F(v)) [since F preserves vector addition]
= G(F(u)) + G(F(v)) [since G preserves vector addition]
= (G ◦ F)(u) + (G ◦ F)(v) [by definition of composition]
Therefore, G ◦ F preserves vector addition.
Now let's consider preservation of scalar multiplication:
(G ◦ F)(ku) = G(F(ku)) [by definition of composition]
= G(kF(u)) [since F preserves scalar multiplication]
= kG(F(u)) [since G preserves scalar multiplication]
= k(G ◦ F)(u) [by definition of composition]
Therefore, G ◦ F preserves scalar multiplication.
Since G ◦ F satisfies both properties of linearity, it is a linear function that preserves vector addition and scalar multiplication.
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a rectangular field has side lenght that measure 9\10 mile and 1\2 mile. what is the area of the field?
The area of rectangular field is [tex]\frac{9}{20} \ \text{miles}^2[/tex].
What is area of rectangle?The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area. In essence, the area of a rectangle is equal to the sum of its length and breadth.
Here the given rectangle , length= [tex]\frac{9}{10}[/tex]miles and Breadth = [tex]\frac{1}{2}[/tex]miles
Now using area of rectangle formula,
=> A = length× breadth square unit.
=> A = [tex]\frac{9}{10}\times\frac{1}{2}[/tex]
=> A = [tex]\frac{9}{20} \ \text{miles}^2[/tex]
Hence the area of rectangular field is [tex]\frac{9}{20} \ \text{miles}^2[/tex].
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The original price of a pair of shorts was £28. In a sale they were reduced by of their original price. What was their sale price? Original price = £28 Sale price = £
Answer:
free
Step-by-step explanation:
28-28=0
Please type the answer by company so that i can see it clearly, thank you!
The occupational safety of workers in Country ABC piqued the curiosity of a safety officer. Country ABC was split into 18 districts by the officer. For personal interviews, five workers were chosen at random from each district. The following are some of the questions that were asked during the interview.
Question (I) – How many days did you work on November 2021 (total 30 days)?
Question (II) – Which district are you living in?
Question (III) – Do you agree that the safety standard in your working environment is high? (Totally disagree/ disagree/ neutral/ agree/ totally agree)
Question (IV) – How much is your daily salary (in HK$100)?
Questions
For each of the following variables, determine whether the variable is qualitative or quantitative. If the variable is quantitative, determine whether the variable is discrete or continuous. In addition, indicate the level of measurement.
(i) The number of days that the worker worked on November 2021
(ii) District that the worker is living in
(iii) Level of agreement on the high safety standard in the working environment of the worker
(iv) Daily salary of the worker (in HK$100)
Quantitative, discrete, ratio
(i) Quantitative, discrete, interval
(ii) Qualitative, nominal
(iii) Qualitative, ordinal
(iv) Quantitative, discrete, ratio
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I need help on 8 and 9!!
Answer:
8) the ordered pair (-1,-1) is a solution
9)the ordered pair (-8,-2) is a solution
Step-by-step explanation:
8) 3x-5y≥2
3(-1)-5(-1)=2
2=2
9)-x-6y>12
-(-8)-6(2)=-4
-4>12
In quadrilateral QRST, QS = RT. Is QRST a rectangle?
Answer:
I believe that no, it is not a rectangle because we don't know if the other 2 sides equal each other. In a rectangle 2 sets of sides are equal so no. it is not a rectangle.
thanks for coming to my ted talk
Step-by-step explanation:
Consider the expression.
17 (5) (9 + 14y)
Select all statements about the expression that are true.
There are exactly 4 terms.
One term of the expression is 23
The expression has exactly 3 factors.
The constant in the factor 9 + 14y is 9.
The factors in the expression are 17.5.9, and 14y.
Answer: None of the statements are true.
The expression has only two terms: 17 and (5)(9 + 14y).
No term in the expression equals 23.
The expression has three factors: 17, 5, and (9 + 14y).
The constant in the factor 9 + 14y is 9.
The factors in the expression are 17, 5, 9, and (9 + 14y).
Step-by-step explanation:
What is the average rate of change of f(X) from x=-3 to x = 6?
f(x) = x4
4 + 4x - 15
Enter your answer in the blank.
PLEASE HELPPP
Given that tangent theta = negative 1, what is the value of secant theta, for StartFraction 3 pi Over 2 EndFraction less-than theta less-than 2 pi?
Negative StartRoot 2 EndRoot
StartRoot 2 EndRoot
0
1
The value of secant theta is √2. The solution has been obtained by using trigonometry.
What is trigonometry?Trigonometry, a subfield of mathematics, is the study of the sides, angles, and connections of the right-angle triangle.
We are given that tangent theta = -1
We know that tan θ = sin θ / cos θ
So,
⇒ -1 = sin θ / cos θ
⇒ -cos θ = sin θ
We know that angle θ lies in the 4th quadrant i.e. between 3π/2 and 2π.
In the 4th quadrant, at 7π/4, the above is true.
So, at θ = 7π/4, we get
cos (7π/4) = √2/2
We know that
sec θ = 1 / cos θ
So,
sec θ = √2
Hence, the value of secant theta is √2.
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The covariance between the returns of A and B is -0. 112. The standard deviation of the rates of return is 0. 26 for stock A and 0. 81 for stock B. The correlation of the rates of return between A and B is closest to: A. )-1. 88 B. )-. 53 C. ). 53 D. )1. 88
The correlation of the rates return between stock A and B for the given covariance and standard deviation is given by option B. -0.53.
Covariance between stock A and B = -0.112
Standard deviation of the rates return for stock A = 0.26
Standard deviation of the rates return for stock B = 0.81
Formula for correlation in terms of covariance and standard deviations is
Correlation = covariance / (standard deviation of A x standard deviation of B)
Here correlation of the rates of return between A and B.
Substitute the given values we get,
⇒ Correlation = (-0.112) / (0.26 x 0.81)
⇒ Correlation ≈ -0.430769 / 0.81
⇒ Correlation ≈ -0.5318
⇒ Correlation ≈ -0.53
Therefore, the correlation of the rates of return between A and B is is equal to option B. -0.53.
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What does a simple machine do when it makes your effort less?
makes work go faster
makes your power more
makes the amount of work you do less
makes resistance force greater than applied force
Answer: it can change the amount of force that you can apply on an object.
Step-by-step explanation:
Answer:
Simple machine makes work go faster
Step-by-step explanation:
If you replace human Labour with simple machine, work is completed faster. It increases output
Compute A² – 2A + I. A= | 1 0 -1| . | 0 -4 0| . |2 0 2| NOTE: Write the elements of the matriz exactly. A2 - 2A +I=
The elements of the matrix are: A² – 2A + I = | 0 0 -1 | | 0 25 0 | | 2 0 -1 |. To compute A² – 2A + I, we need to first find A², then multiply A by 2, and finally add the identity matrix I to the result.
The identity matrix I is a matrix with 1s on the main diagonal and 0s elsewhere.
A² = A * A = | 1 0 -1| * | 1 0 -1| = | 1*1+0*0+(-1)*2 1*0+0*(-4)+(-1)*0 1*(-1)+0*0+(-1)*2 |
| 0 -4 0| | 0 -4 0| | 0*1+(-4)*0+0*2 0*0+(-4)*(-4)+0*0 0*(-1)+(-4)*0+0*2 |
| 2 0 2| | 2 0 2| | 2*1+0*0+2*2 2*0+0*(-4)+2*0 2*(-1)+0*0+2*2 |
= | 1 0 -3 |
| 0 16 0 |
| 6 0 2 |
2A = 2 * | 1 0 -1| = | 2 0 -2|
| 0 -4 0| | 0 -8 0|
| 2 0 2| | 4 0 4|
I = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
A² – 2A + I = | 1 0 -3 | - | 2 0 -2 | + | 1 0 0 | = | 1-2+1 0-0+0 -3-(-2)+0 |
| 0 16 0 | | 0 -8 0 | | 0 1 0 | | 0-0+0 16-(-8)+1 0-0+0 |
| 6 0 2 | | 4 0 4 | | 0 0 1 | | 6-4+0 0-0+0 2-4+1 |
= | 0 0 -1 |
| 0 25 0 |
| 2 0 -1 |
Therefore, A² – 2A + I = | 0 0 -1 |
| 0 25 0 |
| 2 0 -1 |
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Find the distance from the point $\left(1,\ 2\right)$ to the line $y=\frac{1}{2}x-3$
. Round your answer to the nearest tenth.
The distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.
What is expression ?In mathematics, an expression is a combination of numbers, variables, and operators, which when evaluated, produces a value. An expression can contain constants, variables, functions, and mathematical operations such as addition, subtraction, multiplication, and division.
According to given information :To find the distance from a point to a line, we need to find the length of the perpendicular segment from the point to the line.
The line [tex]$y=\frac{1}{2}x-3$[/tex] can be rewritten in slope-intercept form as [tex]$y = \frac{1}{2}x - 3$[/tex], so its slope is [tex]\frac{1}{2}$.[/tex]
A line perpendicular to this line will have a slope that is the negative reciprocal of [tex]\frac{1}{2}$[/tex], which is [tex]-2$.[/tex]
We can then use the point-slope form of a line to find the equation of the perpendicular line that passes through the point [tex]$(1,2)$[/tex]:
[tex]$y - 2 = -2(x - 1)$[/tex]
Simplifying, we get:
[tex]$y = -2x + 4$[/tex]
Now we need to find the point where the two lines intersect, which will be the point on the line [tex]$y = \frac{1}{2}x-3$[/tex] that is closest to [tex]$(1,2)$[/tex]. We can do this by setting the equations of the two lines equal to each other and solving for [tex]$x$[/tex]:
[tex]$\frac{1}{2}x - 3 = -2x + 4$[/tex]
Solving for [tex]$x$[/tex], we get:
[tex]$x = \frac{14}{5}$[/tex]
To find the corresponding [tex]$y$[/tex] value, we can substitute this value of [tex]$x$[/tex] into either of the two line equations. Using [tex]$y = \frac{1}{2}x-3$[/tex], we get:
[tex]$y = \frac{1}{2} \cdot \frac{14}{5} - 3 = -\frac{7}{5}$[/tex]
Therefore, the point on the line [tex]$y = \frac{1}{2}x-3$[/tex] that is closest to [tex]$(1,2)$[/tex] is [tex]$\left(\frac{14}{5}, -\frac{7}{5}\right)$[/tex].
Finally, we can use the distance formula to find the distance between [tex]$(1,2)$[/tex] and [tex]$\left(\frac{14}{5}, -\frac{7}{5}\right)$[/tex]:
[tex]$\sqrt{\left(\frac{14}{5} - 1\right)^2 + \left(-\frac{7}{5} - 2\right)^2} \approx 3.7$[/tex]
Rounding to the nearest tenth, the distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.
Therefore, the distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.
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Si el dividendo es 872 y el cociente es 62 y residuo es 4 cual es el divisor
If the dividend is 872 and the quotient is 62 and remainder is 4, the divisor is equal to 14.
What is a quotient?In Mathematics, a quotient can be defined as a mathematical expression that is typically used for the representation of the division of a number by another number.
How to calculate the dividend?In Mathematics, dividend can be calculated by using this mathematical expression:
Dividend = divisor × quotient + residual
Substituting the given data points into the dividend formula, we have the following;
Dividend = divisor × quotient + residual
872 = divisor × 62 + 4
Divisor = (872 - 4)/62
Divisor = 868/62
Divisor = 14
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Complete Question;
If the dividend is 872 and the quotient is 62 and remainder is 4. What is the divisor?
A person has rupees 210 he wants to donate it he gives rupees 1 for first day rupees 2 for second day rupees 3 for third day and so on for donation he can donate this for maximum how munch days
Answer: This is a classic problem in mathematics known as the arithmetic series or the Gauss sum.
To find out how many days the person can donate with a total of Rs. 210, we need to sum the sequence of donations until we reach the total amount of Rs. 210. The sequence of donations is:
1 + 2 + 3 + 4 + 5 + ... + n
The sum of the sequence can be expressed as:
n(n+1)/2
So we need to solve the equation:
n(n+1)/2 = 210
n(n+1) = 420
n^2 + n - 420 = 0
We can solve this quadratic equation using the quadratic formula:
n = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 1, and c = -420.
n = (-1 ± sqrt(1^2 - 4(1)(-420))) / 2(1)
n = (-1 ± sqrt(1 + 1680)) / 2
n = (-1 ± sqrt(1681)) / 2
n = (-1 ± 41) / 2
Since we are looking for a positive integer value of n, we can discard the negative solution:
n = (41 - 1) / 2
n = 40 / 2
n = 20
Therefore, the person can donate for a maximum of 20 days with a total donation of Rs. 210, starting with Rs. 1 on the first day and increasing by Rs. 1 each day.
Step-by-step explanation:
what’s the answer lol <3
Answer:
Should be the second answer.
Step-by-step explanation:
Consider the two functions x1 = y^2 and x2 = b on the y-interval [-a, a], if a = 6.2. What value does b have to be in order for the area between x, and x2 and [-a, a], to equal 50.5? Round your answer to five decimal places.
The value of b that makes the area between the two functions equal to 50.5 on the interval [-6.2, 6.2] is 29.64355.
To find the value of b that makes the area between the two functions equal to 50.5 on the interval [-6.2, 6.2], we need to set up an integral and solve for b.
First, we need to find the difference between the two functions:
x2 - x1 = b - y^2
Next, we need to integrate this difference over the given interval:
∫[-6.2, 6.2] (b - y^2) dy
Using the power rule for integration, we get:
[b*y - (y^3)/3] from -6.2 to 6.2
Plugging in the values for the interval and simplifying, we get:
(6.2b - 158.488) - (-6.2b - 158.488)
Simplifying further, we get:
12.4b - 316.976
Now, we can set this equal to the given area and solve for b:
12.4b - 316.976 = 50.5
12.4b = 367.476
b = 367.476/12.4
b = 29.64354839
Rounding to five decimal places, we get:
b = 29.64355
So, the value of b that makes the area between the two functions equal to 50.5 on the interval [-6.2, 6.2] is 29.64355.
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3. Letfbe a differentiable function of one variable, andw=f(ex+2xy). (a) Verify that the functionwsatisfies the equation(ex+2y)wy−2xwx=0. (b) Iff(u)=cosu, calculatewxy. 4. Consider the functionf(P,t)=f(x(t),y(t),z(t),t)=t(ysinx+ez)and the pointP=(π,1,0). Find the material derivative offatP
3 a. The function w satisfies the equation (ex+2y)wy−2xwx=0.
3 b. The value of wxy is -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)
4 . The material derivative of fatP is 0.
3. (a) To verify that the function w satisfies the equation (ex+2y)wy−2xwx=0, we can take the partial derivatives of w with respect to x and y, and then substitute them into the equation.
First, let's find the partial derivatives of w:
∂w/∂x = f'(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x)
Now, we can substitute these partial derivatives into the equation:
(ex+2y)(f'(ex+2xy)(2x)) - 2x(f'(ex+2xy)(ex+2y)) = 0
Simplifying this equation, we get:
2exf'(ex+2xy) + 4xyf'(ex+2xy) - 2exf'(ex+2xy) - 4xyf'(ex+2xy) = 0
This simplifies to 0 = 0, which is true. Therefore, the function w satisfies the equation (ex+2y)wy−2xwx=0.
(b) If f(u) = cosu, then we can find wxy by taking the partial derivative of w with respect to x and y, and then substituting f(u) = cosu:
∂w/∂x = f'(ex+2xy)(ex+2y) = -sin(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x) = -sin(ex+2xy)(2x)
Now, we can find wxy by taking the partial derivative of ∂w/∂x with respect to y:
wxy = ∂(∂w/∂x)/∂y = ∂(-sin(ex+2xy)(ex+2y))/∂y = -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)
4. To find the material derivative of fatP, we can use the formula:
Df/Dt = ∂f/∂t + ∂f/∂x(dx/dt) + ∂f/∂y(dy/dt) + ∂f/∂z(dz/dt)
First, let's find the partial derivatives of f:
∂f/∂t = ysinx + ez
∂f/∂x = ty*cosx
∂f/∂y = tsinx
∂f/∂z = tez
Now, we can find the material derivative of fatP by substituting the point P = (π,1,0) and the derivatives of x, y, and z with respect to t:
Df/Dt = (1*sinπ + e^0) + (π*1*cosπ)(dx/dt) + (π*sinπ)(dy/dt) + (π*e^0)(dz/dt) = 0 + 0 + 0 + 0 = 0
Therefore, the material derivative of fatP is 0.
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ach in the lowest-yielding, least-risky acc uld she invest in each account to achieve x+y+z=50,000 0.03x+0.055y+0.09z=2540
The investor should invest $32,500 in the lowest-yielding, least-risky account, $20,000 in the medium-yielding, medium-risk account, and $17,500 in the highest-yielding, highest-risk account.
To solve this problem, we can use a system of linear equations.
We have three equations and three unknowns: x, y, and z.
The equations are: x + y + z = 50,0000.03x + 0.055y + 0.09z = 2540
We can use substitution or elimination to solve for one of the variables and then plug that value back into the other equations to find the remaining variables.
For example, we can solve for x in the first equation:
x = 50,000 - y - z
Then we can substitute this value of x into the second equation:
0.03(50,000 - y - z) + 0.055y + 0.09z = 2540
Simplifying this equation gives us:
1500 - 0.03y - 0.03z + 0.055y + 0.09z = 25400.025y + 0.06z = 1040
Now we can solve for one of the remaining variables, such as y:
y = (1040 - 0.06z) / 0.025
And we can substitute this value of y back into the first equation to find z:
50,000 - (1040 - 0.06z) / 0.025 - z = 50,000
Solving for z gives us:
z = 17,500
Finally, we can plug this value of z back into the equations for x and y to find the remaining variables:
x = 50,000 - y - 17,500 = 32,500 - y
y = (1040 - 0.06(17,500)) / 0.025 = 20,000
So the solution is x = 32,500, y = 20,000, and z = 17,500.
This means that the investor should invest $32,500 in the lowest-yielding, least-risky account, $20,000 in the medium-yielding, medium-risk account, and $17,500 in the highest-yielding, highest-risk account.
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The value of x is decreased by 74% . Which expression represents this situation?
74x
0.74x
0.926x
0.26x
Answer:
0.26x
Step-by-step explanation:
If x is decreased by 74% then new value of x
= x - 74% of x
74% as decimal = 74/100 = 0.74
New value of x
= x - 0.74x
=x(1 - 0.74)
=0.26x
Suppose x > y > 0 and a > b > 0. Is it true that x/b > y/a? if so
For the inequality x > y > 0 and a > b > 0 the expression ( x /y)> (x +b)/ (y+ a) > b/a is true if x/b > y/a.
For x > y > 0 and a > b > 0
The inequality x/b > y/a
Simplify by cross-multiplication we get,
⇒xa > yb
Adding xy to both sides,
⇒xy + xa > xy + yb
Factoring the left-hand side,
⇒ x(y + a) > y(x + b)
Dividing both sides by (y + a)(x + b), as x > y > 0 and a > b > 0,
⇒ x/(x + b) > y/(y + a)
Multiplying both sides by x/y we get the expression,
⇒x/y > (x + b)/(y + a) __(1)
It proves the half part of the expression, x/y > (x + b)/(y + a)
Now second part x/y > (x + b)/(y + a) > b/a.
Using inequality x/b > y/a to get:
a/b > y/x
Multiplying both sides by (x + b)/(y + a),
⇒(a/b) × (x + b)/(y + a) >(y/x) × (x + b)/(y + a)
Expand both sides and simplifying,
⇒ ( ax + ab ) / (by + ab ) > ( xy + by ) / ( xy + ax )
⇒( ax + ab )( xy + ax ) > ( xy + by ) (by + ab )
⇒ ax²y + a²x² + abxy + a²bx > by²x + abxy + b²y² + ab²y
⇒ (ax -by )( x + b )( y + z) > 0
⇒ax - by > 0 or ( x + b )> 0 or ( y + z) > 0
⇒ ax > by
⇒ x /y > b /a
As a > b > 0
⇒ (x +b)/ (y+ a) > b/a __(2)
From (1) and (2) we have,
( x /y)> (x +b)/ (y+ a) > b/a
Therefore , the expression ( x /y)> (x +b)/ (y+ a) > b/a is true for the given condition.
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The above question is incomplete, the complete question is:
Suppose x > y > 0 and a > b > 0. Is it true that x/b > y/a then expression
( x /y)> (x +b)/ (y+ a) > b/a.
Use the special factoring methods to factor the following binomial. If it cannot be factored, indicate "Not Factorable". 121y^(8)z^(2)-256x^(6)
The binomial 121y8z2 - 256x6 can be factored using the difference of two squares rule.
To factor the binomial 121y8z2 - 256x6, use the special factoring methods. Notice that the binomial has a difference of two squares, where the first term is a perfect square and the second term is the square of a binomial. This means that you can factor this binomial using the difference of two squares rule:
[tex]121y8z^2 - 256x^6 = (11y4z^2 - 16x^3) * (11y4z^2 + 16x^3)[/tex]
Therefore, the binomial 121y8z2 - 256x6 can be factored using the difference of two squares rule.
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How do I solve these two ?
Geometry
The length of the sides x and y of the right-angled triangles using the trigonometric ratio of sine and cosine are:
11). x = 11√3 and y = 33
12). x = 4√7 and y = 8√7
What is trigonometric ratios?The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.
The basic trigonometric ratios includes;
sine, cosine and tangent.
recall that sin60° = √3/2 and cos60° = 1/2
For question 11:
sin 60° = y/22√3 {opposite/hypotenuse}
y = 22√3 × sin 60° {cross multiplication}
y = 22√3 × √3/2
y = 11 × 3
y = 33
cos 60° = x/22√3 {adjacent/hypotenuse}
x = 22√3 × cos 60° {cross multiplication}
x = 22√3 × 1/2
x = 11√3.
For question 12:
sin 60° = 4√21/y {opposite/hypotenuse}
y = 4√21/sin 60° {cross multiplication}
y = 4√21 ÷ √3/2
y = 4√21 × 2/√3
y = 4√7 × 2
y = 8√7
cos 60° = x/8√7 {adjacent/hypotenuse}
x = 8√7 × cos 60° {cross multiplication}
x = 8√7 × 1/2
x = 4√7
Therefore, the length of the sides x and y of the right-angled triangles using the trigonometric ratio of sine and cosine are:
11). x = 11√3 and y = 33
12). x = 4√7 and y = 8√7
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For the points (−8,5) and (−6,−1), (a) Find the exact distance between the points. (b) Find the midpoint of the line segment whose endpoints are the given points. Part 1 of 2 (a) The exact distance between the points is Part 2 of 2 (b) The midpoint is
(a) The exact distance between the points is 2√10.
(b) The midpoint of the line segment is (-7, 2).
(a) To find the distance between the two given points, use the distance formula, which is:
d = √[(x₂ - x₁)² + (y₂ - y₁)²].
In this case, x₁ = -8, y₁ = 5, x₂ = -6, and y₂ = -1.
Plugging these values into the formula gives us:
d = √[(-6 - (-8))² + (-1 - 5)²]
d = √[(2)² + (-6)²]
d = √[4 + 36]
d = √40
d = 2√10
(b) The midpoint of a line segment can be found using the midpoint formula, which is:
(x₁ + x₂)/2, (y₁ + y₂)/2.
In this case, x₁ = -8, y₁ = 5, x₂ = -6, and y₂ = -1.
Plugging these values into the formula gives us:
midpoint = [(-8 + -6)/2, (5 + -1)/2]
midpoint = [(-14)/2, (4)/2]
midpoint = (-7, 2)
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