Answer:
See below.
Step-by-step explanation:
5/6 is a fraction. The 5 is in the numerator, and the 6 is in the denominator.
The denominator is the number of parts the unit was divided into. In this case, the denominator is 6. That means one unit, 1, was divided into 6 equal parts. Each part is one-sixth.
The numerator is the number of those parts that you use. 5 in the numerator means to use 5 of those parts, each of which is 1/6 of 1.
In other words, 5/6 means divide 1 into 6 equal parts, and take 5 of those parts.
5/6 is the same as 5 divided by 6, so as a decimal it is 0.8333...
As a percent it is 83.333...%
Forty ounces of Lazy Lawn fertilizer covers 1,250 square feet of lawn.
(a) How many ounces would be required to cover a 6,000 square foot lawn?
(b) If Lazy Lawn costs $1.17 for a 24 ounce bag, what is the total cost (in dollars) to fertilize the lawn?
$
To cover a 6,000 square foot lawn, 192 ounces of fertilizer will be required. With the given cost of the fertilizer, the total cost to fertilize the lawn would be $9.36.
Explanation:(a) To calculate how many ounces are required to cover a 6,000 square foot lawn, we can set up a proportion.
40 ounces of Lazy Lawn fertilizer covers 1,250 square feet, so let's represent it as 40/1250. We know that we have a 6,000 square foot lawn but we don't know how many ounces it requires, let's represent it as x/6000.
Our proportion will then be as follows: 40/1250 = x/6000. Cross multiply to find 'x', we will get x = [(40*6000)/1250] = 192 ounces.
(b) Now, to calculate the total cost we first need to know how many 24 ounce bags we will need. 192 ounces divided by 24 ounces per bag gives us 8 bags. Then we calculate the cost, 8 bags times $1.17 per bag gives us a total of $9.36.
Learn more about Proportional problems here:https://brainly.com/question/32581317
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Let A = {a, b, c, d, e} and B = {a, c, f, g, i}. Universal Set: ∪= {a,b,c,d,e,f,g,h,i}
1. A ∪ B^c
2. B - A
Answer:
1. { a, b, c, d, e, h }
2. { f, g, i }
Step-by-step explanation:
Given sets,
A = {a, b, c, d, e},
B = {a, c, f, g, i}
Universal set , ∪ = {a, b, c, d, e, f, g, h, i},
1. Since, [tex]B^c[/tex] = elements of universal set which are not in set B
= U - B
= { b, d, e, h },
Thus,
[tex]A\cup B^c[/tex] = All elements of A and [tex]B^c[/tex]
= { a, b, c, d, e, h }
2. B - A = elements of set B which are not in set A
= { f, g, i }
If $14,000 is invested at 4% compounded quarterly, what is the amount after 8 years?
The amount after 8 years will be
Answer:
The amount after 8 years is $19249.17
Step-by-step explanation:
For any calculation for investments there si the compound interest formula:
[tex]A=P(1+\frac{r}{n} )^(n*t)[/tex]
Where
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year
So for this example
P, the original amount ($14000)
r, 4%
t, 8 years
A, the amount after 8 years
n, 4, due that is quarterly
[tex]P=$14000(1+((4/100)/(4)))^(4*8)\\\\P= $19249.17[/tex]
Use induction to prove that 2? ?? for any integer n>0 . Indicate type of induction used.
I proved the base case using n = 1, and for my induction hypothesis, I said that we assume n = k for 2^k > k, but I am stuck trying to get to n = k + 1.
So far I have:
2^k > k
2*2^k > 2*k
2^{k+1} > 2k
Answer with explanation:
The given statement is which we have to prove by the principal of Mathematical Induction
[tex]2^{n}>n[/tex]
1.→For, n=1
L H S =2
R H S=1
2>1
L H S> R H S
So,the Statement is true for , n=1.
2.⇒Let the statement is true for, n=k.
[tex]2^{k}>k[/tex]
---------------------------------------(1)
3⇒Now, we will prove that the mathematical statement is true for, n=k+1.
[tex]\rightarrow 2^{k+1}>k+1\\\\L H S=\rightarrow 2^{k+1}=2^{k}\times 2\\\\\text{Using 1}\\\\2^{k}>k\\\\\text{Multiplying both sides by 2}\\\\2^{k+1}>2k\\\\As, 2 k=k+k,\text{Which will be always greater than }k+1.\\\\\rightarrow 2 k>k+1\\\\\rightarrow2^{k+1}>k+1[/tex]
Hence it is true for, n=k+1.
So,we have proved the statement with the help of mathematical Induction, which is
[tex]2^{k}>k[/tex]
Increasing at a constant rate,a company's profits y have gone form $535 milion in 1985 to $570 million in 1990. Find the expected level of profit for 1995 if the trend continues. 2)
Answer:
total profit=$607.278
Step-by-step explanation:
company's profit in 1985= $535 million
company's profit in 1990=$570 million
growth rate = [tex]\frac{570-535}{535}\times 100[/tex]
= [tex]\frac{35}{535} \times 100[/tex]
= 6.54 %
profit in year 1995 will be = [tex]\frac{6.54}{100}\times 570 =\ \$37.278[/tex]
hence total profit= $570+$37.278
= $607.278
In order to compare two scales, 30 objects are weighed on both scales. Each object would then have two weight values (one from scale 1 and one from scale 2). Based on the nature of the differences in the two weight measurements for the 30 objects, the two scales may be compared. Do these samples represent dependent or independent samples? A. dependent samples. B. independent samples.
Answer:
These two samples are independent samples.
Step-by-step explanation:
Each object would then have two weight values (one from scale 1 and one from scale 2). Based on the nature of the differences in the two weight measurements for the 30 objects, the two scales may be compared.
These two samples are independent samples.
Though the objects are same but the scales are different.
Deines Corporation has fixed costs of $480,000. It has a unit selling price of $6, unit variable cost of $4.4, and a target net income of $1,500,000. Compute the required sales in units to achieve its target net income.
Answer:
The required sales in units to achieve its target net income is 1,237,500 units.
Step-by-step explanation:
From the given information it is clear that
Fixed cost = $480,000
Selling Price = $6 per unit
Variable Cost = $4.4 per unit
Target net income = $1,500,000
We need to find the required sales in units to achieve its target net income.
[tex]Units=\frac{\text{Fixed cost + Target net income}}{\text{Selling Price - Variable Cost}}[/tex]
[tex]Units=\frac{480000+1500000}{6-4.4}[/tex]
[tex]Units=\frac{1980000}{1.6}[/tex]
[tex]Units=1237500[/tex]
Therefore the required sales in units to achieve its target net income is 1,237,500 units.
Consider the random variables X and Y with joint density function ???? f(x,y)= x+y, 0≤x≤1;0≤y≤1 0, elsewhere. (a) Find the marginal distributions of X and Y . (b) Find P(X > 0.25,Y > 0.5).
a. The marginal densities
[tex]f_X(x)=\displaystyle\int_0^1(x+y)\,\mathrm dy=x+\frac12[/tex]
and
[tex]f_Y(y)=\displaystyle\int_0^1(x+y)\,\mathrm dx=y+\frac12[/tex]
b. This can be obtained by integrating the joint density over [0.25, 1] x [0.5, 1]:
[tex]P(X>0.25,Y>0.5)=\displaystyle\int_{1/4}^1\int_{1/2}^1(x+y)\,\mathrm dx\,\mathrm dy=\frac{33}{64}[/tex]
Final answer:
To find the marginal distributions of X and Y, we integrate the joint density function over the range of the other variable. The marginal distribution of X is f(x) = x+1/2, for 0≤x≤1. The marginal distribution of Y is f(y) = y+1/2, for 0≤y≤1.
Explanation:
To find the marginal distributions of X and Y, we need to integrate the joint density function over the range of the other variable. For the marginal distribution of X, we integrate f(x,y) with respect to y from 0 to 1:
(∫⁰₁(x+y) dy) = (x+y/2)∣⁰₁ = x+1/2
So, the marginal distribution of X is given by f(x) = x+1/2, for 0≤x≤1.
Similarly, for the marginal distribution of Y, we integrate f(x,y) with respect to x from 0 to 1:
(∫⁰₁(x+y) dx) = (x2/2+xy)∣⁰₁ = y+1/2
Therefore, the marginal distribution of Y is given by f(y) = y+1/2, for 0≤y≤1.
A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 11 − x 2 . What are the dimensions of such a rectangle with the greatest possible area?
Answer:[tex]\frac{22}{3}[/tex],[tex]2\dot \sqrt{\frac{11}{3}}[/tex]
Step-by-step explanation:
Given
rectangle with its base on x-axis
and other two corners at parabola
and parabola is downward facing symmetric about y-axis
let y be the y co-ordinate of the corner thus x co-ordinate is given by
[tex]x=\pm \sqrt{11-y}[/tex]
Thus lengths of rectangle is [tex]2\sqrt{11-y}[/tex] & y
Area [tex]=y\times 2\sqrt{11-y}[/tex]
differentiating w.r.t to y for maximum area
[tex]\frac{\mathrm{d} A}{\mathrm{d} y}=2\times \sqrt{11-y}-\frac{y}{2\dot \sqrt{11-y}}=0[/tex]
we get y=[tex]\frac{22}{3}[/tex]
and [tex]x=\pm \sqrt{\frac{11}{3}}[/tex]
A_{max}=16.21 units
The arithmetic mean of any two nonnegative real numbers a and b is greater than or equal to their geometric mean vab. [Hint: consider (Va - vb) 0.]
Answer with explanation:
Here, a and b are two real numbers.
Arithmetic Mean of a and b
[tex]A=\frac{a+b}{2}[/tex]
[tex]\rightarrow a< \frac{a+b}{2}<b[/tex]
Geometric Mean of a and b
[tex]G=\sqrt{ab}[/tex]
[tex]\rightarrow a< \sqrt{ab}<b[/tex]
[tex]A-G\\\\=\frac{a+b}{2}-\sqrt{ab}\\\\=\frac{a+b-2\sqrt{ab}}{2}\\\\=[\frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}}]^2>0\\\\A-G>0\\\\A>G[/tex]
Square of difference of any two numbers is greater than or equal to 0.
⇒A.M of two Numbers > G.M of two Numbers
Six years ago my son was one-third my age at that time.
Six years from now he will be one-half my age at that time.
How old is my son?
m - my age
s - son's age
[tex]s-6=\dfrac{m-6}{3}\\s+6=\dfrac{m+6}{2}\\\\3s-18=m-6\\2s+12=m+6\\\\m=3s-12\\m=2s+6\\\\3s-12=2s+6\\s=18[/tex]
He's 18
9. An RSA cryptosystem has modulus n 391, which is a product of the primes 23 and 17. Which of the following is suitable as an encoding key e? (a) 163 (b) 353 (c) 351 (d) 277 (e) none of these. 10. Which of the following polynomials p(x) is complete over Zalr? (a) z4+1 (e) none of these
Answer:
163
Step-by-step explanation:
So n=391.
This means p=23 and q=17 where p*q=n.
[tex] \lambda (391)=lcm(23-1,17-1)=lcm(22,16)=2*8*11=16*11=176. [/tex]
We want to choose e so that e is between 1 and 176 and the gcd(e,176)=1.
There is only one number in your list that is between 1 and 176... Hopefully the gcd(163,176)=1.
It does. See notes below for checking it:
176=2(88)=2(4*22)=2(2)(2)(2)(11)
None of the prime factors of 176 divide 163 so we are good.
The answer is 163.
John took all his money from his savings account. He spent $110 on a radio and 4/11 of what was left on presents for his friends. John then put 2/5 of his remaining money into a checking account and donated the $420 that was left to charity. How much money did John originally have in his savings account?
Answer:
$1210
Step-by-step explanation:
Let x be total amount
First John spent $110 on a radio and 4/11 of what was left on presents for his friends so he was left with
[tex]\frac{7}{11}(x-110)=\frac{7}{11}x-70[/tex]
Then he put 2/5 of his remaining money into a checking account
[tex]\frac{3}{5}\left(\frac{7}{11}x-70\right)[/tex]
Rest he donated to charity
[tex]420=\frac{3}{5}\left(\frac{7}{11}x-70\right)\\\Rightarrow \left(\frac{7}{11}x-70\right)=\frac{2100}{3}\\\Rightarrow \frac{7}{11}x-70=700\\\Rightarrow \frac{7}{11}x=770\\\Rightarrow x=1210[/tex]
Hence total amount of money John originally had was $1210
74% of workers got their job through college. Express the null and alternative hypotheses in symbolic form for this claim (enter as a decimal WITH a leading zero: example 0.31)
Answer: Null hypothesis = [tex]H_0:p=0.74[/tex]
Alternative hypothesis = [tex]H_1:p\neq0.24[/tex]
Step-by-step explanation:
Given claim : 74% of workers got their job through college.
In proportion , 0.74 of workers got their job through college.
Let p be the proportion of workers got their job through college.
Then claim : [tex]p=0.74[/tex]
We know that the null hypothesis always takes equality sign and alternative hypothesis takes just opposite of the null hypothesis.
Thus, Null hypothesis = [tex]H_0:p=0.74[/tex]
Alternative hypothesis = [tex]H_1:p\neq0.24[/tex]
Discount on LCD TV is $240.
Sale Price is $1575.00
What was the list price?
Answer: The list price was $1815.00.
Step-by-step explanation: Given that the discount on a LCD TV is $240 and the sale price is $1575.00.
We are to find the list price.
The discount is given on the price that is listen on the LCD TV.
So, the list price will be equal to the sum of the sale price and the discount price.
Therefore, the required list price of the LCD TV is given by
[tex]L.P.\\\\=\textup{sale price}+\textup{discount}\\\\=\$(1575.00+240.00)\\\\=\$1815.00.[/tex]
Thus, the list price was $1815.00.
Suppose that you currently own a clothes dryer that costs $25 per month to operate A new efficient dryer costs $630 and has an estimated operating cost of $15 per month. How long will it take for the new dryer to pay for itself? months The clothes dryer will pay for itself in
Answer:
Dryer will pay for itself in 63 months or 5 years and 3 months.
Step-by-step explanation:
Let after x months new dryer will pay for itself.
Old dryer is costing $25 to operate so after x months it will cost = 25x
Similarly new dryer which cost $630 and operating cost is $15 per month.
So after x months new drier will cost = $(630 + 15x)
If the new dryer pay for itself in x months then total cost of both the dryers after x months should be same.
Therefore, 25x = 630 + 15x
25x - 15x = 630
10x = 630
x = [tex]\frac{630}{10}[/tex]
x = 63 months
Or x = 5 years 3 months
Answer is 63 months or 5 years 3 months.
40% of the groups budget was spent on activities. If $64,000 was spent on the activities, what was the full budget? solve with the aid of a diagram. State what type of problem this is. if it's division, which type? Explain your reasoning, not just by using an equation.
Answer:
The full budget has a total value of $160,000
Step-by-step explanation:
This is a simple ratio problem and can be solved using the Rule of Three property. The Rule of Three is used to compare two ratios and find the missing part. In this case the ratios would be the following.
[tex]\frac{64,000}{40 percent} = \frac{x}{100 percent}[/tex]
[tex]\frac{64,000 * 100 percent}{40 percent} = \frac{x}[/tex]
[tex]\frac{6,400,000}{40 percent} = \frac{x}[/tex]
[tex]160,000= \frac{x}[/tex]
So now we know that 100% (The full budget) has a total value of $160,000.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
A plumbing supply company has fixed costs of $9,000 per month and average variable costs of $9.30 per unit manufactured. The company has $90,000 available to cover the monthly costs. How many units can the company manufacture? (Fixed costs are those that occur regardless of the level of production. Variable costs depend on the level of production. Your answer should be in terms of whole units produced.)
Answer:
8710 units
Step-by-step explanation:
Step 1: Write all the data
Fixed cost: $9000
Average variable cost: 9.3 per unit
Total cost: 90,000
Total units: x
Step 2: Find the total variable cost
Average variable cost is per unit so it has to be multiplied by the number of units to find the total variable cost.
Total variable cost = average variable cost per unit x number of units
Total variable cost = 9.3x
Step 3: Make the formula for finding x
Total cost = total fixed cost + total variable cost
90,000 = 9000 + 9.3x
81000 = 9.3x
x = 8709.67
Rounded off to 8710 units
!!
Recall the formula for finding the area of a rectangle. Define a
variable for the width and set up an equation to find the dimensions of a
rectangle that has an area 144 square inches, given that the length is 10
inches longer than its width.
Final answer:
To solve for the width of the rectangle, define the width as w, set up the equation 144 = w(w + 10), and factor the resulting quadratic equation to find w = 8 inches. Hence, the rectangle's dimensions are 8 inches in width and 18 inches in length.
Explanation:
To find the dimensions of a rectangle with an area of 144 square inches where the length is 10 inches longer than its width, we first recall the formula for the area of a rectangle:
Area = Length × Width
Let's define the width as w, and since the length is 10 inches longer, we can say the length is w + 10. Plugging these into the area formula we get:
144 = w × (w + 10)
Now, we have a quadratic equation to solve for w:
Expand the equation: 144 = w2 + 10w
Subtract 144 from both sides to set the equation to zero: w2 + 10w - 144 = 0
Factor the quadratic equation: (w + 18)(w - 8) = 0
Solve for w: w = -18 or w = 8 (since width cannot be negative, w = 8 is the solution.)
Therefore, the dimensions of the rectangle are a width of 8 inches and a length of 18 inches (8 + 10).
Explain how simulation is used in the real world. Provide a specific example from your own line of work, or a line of work that you find particularly interesting.
Answer:
Explained
Step-by-step explanation:
Simulation is nothing but an approximate or somewhat accurate imitation of a real world situation. Simulation is actually a computer generated graphics to predict how a system will behave under given set of parameters, without actually applying real resources. Simulation finds a variety of application in various fields. Simulation of blood flowing through veins and arteries. Simulation of LBW decisions in a cricket match, which helps Umpires to make correct LBW decisions in a match. A lot of recondite process can understood using Simulation videos that is why concept of smart learning as been introduced.
Find the first 2 terms of each of two power series solutions: y" + x^2 y'+ xy = 0
Apparently this solution is too long for posting, so I've written it elsewhere and am attaching screenshots of it.
The first four terms of each solution are
[tex]1-\dfrac23x^3+\dfrac5{36}x^6-\dfrac{10}{567}x^9[/tex]
and
[tex]x-\dfrac1{24}x^4+\dfrac1{315}x^7-\dfrac7{32,400}x^{10}[/tex]
The graph is a transformation of one of the basic functions. Find the equation that defines the function.
Answer:
So anyways the equation appears to be [tex]y=(x+4)^3+3[/tex].
Step-by-step explanation:
It looks like a cubic to me.
That is the parent function looks like [tex]f(x)=x^3[/tex].
I'm going to identify the transformations here by using the zero of the function I called f. So where has the point (0,0) on [tex]y=x^3[/tex] wonder to in the new graph. It appears to be (-4,3). So the graph moved left 4 units and up 3 units.
f(x+4)+3 moves the graph left 4 and up 3.
----------Other notes:
f(x-4)+3 moves the graph right 4 and up 3.
f(x+4)-3 moves the graph left 4 and down 3.
f(x-4)-3 moves the graph right 4 and down 3.
So anyways the equation appears to be [tex]y=(x+4)^3+3[/tex].
To determine the equation of a transformed basic function, we typically look at how the graph's shape compares to one of the standard basic functions. The basic functions include linear functions, quadratic functions, absolute value functions, square root functions, cubic functions, and exponential and logarithmic functions. We also consider basic trigonometry functions for periodic graphs.
To find the equation, we need to identify four main transformations that may have been applied to the basic function:
1. **Vertical stretching/shrinking**: If the graph is stretched or shrunk vertically, this is represented by a multiplication factor `a` in front of the basic function `f(x)`.
2. **Horizontal stretching/shrinking**: If the graph is stretched or shrunk horizontally, this is represented by a factor within the function's argument, such as `f(bx)`, where `1/b` is the stretching/shrinking factor.
3. **Vertical shifting**: If the graph is shifted up or down, a constant `c` is added or subtracted from the function, giving `f(x) + c`.
4. **Horizontal shifting**: If the graph is shifted left or right, the function's input is adjusted by adding or subtracting a constant `d` within the argument of the function, yielding `f(x - d)`.
5. **Reflections**: If the graph is flipped over the x-axis, this is represented by a negative sign in front of the `a` factor. If it's flipped over the y-axis, the negative sign is inside the function's argument, `f(-x)`.
Without any specific details about the graph's appearance, the points, or what basic function it resembles, it is impossible to provide the exact transformed function. However, for illustrative purposes, I’ll demonstrate how one might find the equation for a transformed quadratic function based on hypothetical graph observations:
Suppose you find that the graph looks like a parabola that opens upwards and has been:
- Stretched vertically by a factor of 3 (vertical stretch)
- Compressed horizontally by a factor of 1/2 (horizontal stretch)
- Shifted up by 5 units (vertical shift)
- Shifted to the right by 4 units (horizontal shift)
With these observations, you would start with the standard quadratic function, `f(x) = x^2`, and apply the transformations:
1. Vertical stretch by 3: `f(x) = 3x^2`
2. Horizontal compression by a factor of 1/2, which is equivalent to stretching by a factor of 2: `f(x) = 3(x/2)^2 = 3(x^2/4) = (3/4)x^2`
3. Vertical shift up by 5 units: `f(x) = (3/4)x^2 + 5`
4. Horizontal shift right by 4 units: `f(x) = (3/4)(x - 4)^2 + 5`
The transformed function based on the hypothetical scenario would be `f(x) = (3/4)(x - 4)^2 + 5`.
Without specifics of the graph in question, you would follow a similar process: identify the basic function type based on the shape of the graph and apply the relevant transformations.
You borrow $680 from your brother and agree to pay back $750 in 3 months. What simple interest rate will you pay?
Answer:
hence rate interest r = 41.176%
Step-by-step explanation:
The amount borrowed= $680
amount payed back = $750
therefore, interest incurred = 750-680= $70
time, t= 3 months = 3/12= 0.25 years
rate%, r
we know that SI= [tex]\frac{PRT}{100}[/tex]
70= [tex]\frac{680\timesr\0.25}{100}[/tex]
r=[tex]\frac{7000}{0.25\times680} = 41.176[/tex]
hence rate interest r = 41.176%
625 ÷ 62.5 × 30 ÷ 10
Answer:
30
Step-by-step explanation:
Follow the correct order of operations.
There are only multiplications and divisions, so do them in the order they appear from left to right.
625 ÷ 62.5 × 30 ÷ 10 =
= 10 × 30 ÷ 10
= 300 ÷ 10
= 30
Samantha is trying to complete the Free Throw wellness challenge. In order to earn her chip, she needs to hit 13 out of 20 free throws on the basketball court Her last three attempts were 12 out of 20, 10 out of 20, and 9 out of 20 How far is her average free throw percentage from the needed free throw percentage to earn the chip? (round to the nearest whole number)
Answer:
21 % below what she needs
Step-by-step explanation:
She had hit 12, 10 and 9
This averages to
(12+10+9)/3 = 31/3 =10 1/3
She needs 13
Percentage = (needed-actual)/needed * 100%
= (13-10 1/3) / 13 * 100%
= (2 2/3) /13 * 100%
=.205128205 * 100%
=20.5128205%
To the nearest whole number
21 %
She is 21 % below what she needs
Samantha's current average free throw percentage is 15 percentage points away from the 65% needed to earn her wellness challenge chip.
Explanation:Samantha is working on improving her free throw percentage in basketball, and she needs to calculate how far her average is from the required target to earn her wellness challenge chip. To determine her average free throw percentage, we first calculate her average number of successful shots by adding her last three attempts and dividing by three: (12 + 10 + 9) / 3 = 31 / 3 = 10.33, rounded to 10 successful shots on average. Since she takes 20 shots each time, her average percentage is (10 / 20) * 100 = 50%.
To earn her chip, she needs to hit 13 out of 20 free throws, which is (13 / 20) * 100 = 65%. The difference between her current average and the needed percentage is 65% - 50% = 15%. Rounding to the nearest whole number, she is 15 percentage points away from the required free throw percentage to succeed in the challenge.
In a certain country, the true probability of a baby being a girl is 0.469. Among the next seven randomly selected births in the country, what is the probability that at least one of them is a boy?
Answer:
The probability is 0.995 ( approx ).
Step-by-step explanation:
Let X represents the event of baby girl,
The probability of a baby being a girl is, p = 0.469,
So, the probability of a baby who is not a girl is, q = 1 - 0.469 = 0.531,
Also, the total number of experiment, n = 7
Thus, by the binomial distribution formula,
[tex]P(x)=^nC_x(p)^x q^{n-x}[/tex]
Where, [tex]^nC_x=\frac{n!}{x!(n-x)!}[/tex]
The probability that all babies are girl or there is no baby boy,
[tex]P(X=7)=^7C_7(0.469)^7(0.531)^{7-7}[/tex]
[tex]=0.00499125661758[/tex]
Hence, the probability that at least one of them is a boy = 1 - P(X=7)
= 1 - 0.00499125661758
= 0.995008743382
≈ 0.995
Questions (no partial grades if you don't show your work) 1. In a group of 6 boys and 4 girls, four children are to be selected. In how many diffeest weys ces they be selected if at least one boy must be there
Answer:
Total number of ways will be 209
Step-by-step explanation:
There are 6 boys and 4 girls in a group and 4 children are to be selected.
We have to find the number of ways that 4 children can be selected if at least one boy must be in the group of 4.
So the groups can be arranged as
(1 Boy + 3 girls), (2 Boy + 2 girls), (3 Boys + 1 girl), (4 boys)
Now we will find the combinations in which these arrangements can be done.
1 Boy and 3 girls = [tex]^{6}C_{1}\times^{4}C_{3}=6\times4[/tex]=24
2 Boy and 2 girls=[tex]^{6}C_{2}\times^{4}C_{2}=\frac{6!}{4!\times2!}\times\frac{4!}{2!\times2!}=15\times6=90[/tex]
3 Boys and 1 girl = [tex]^{6}C_{3}\times^{4}C_{1}=\frac{6!}{4!\times2!}\times\frac{4!}{3!}=\frac{6\times5\times4}{3 \times2} \times4=80[/tex]
4 Boys = [tex]^{6}C_{4}=\frac{6!}{4!\times2!} =\frac{6\times 5}{2\times1}=15[/tex]
Now total number of ways = 24 + 90 + 80 + 15 = 209
Determine the exact formula for the following discrete models:
2tn+2 = 3tn+1 + 2tn; t0 = 1; t1 = 3;
49yn+2 = -16yn; y0 = 0; y1 = 2;
9xn+2 = 12xn+1- 85xn; x0 = 0; x1 =1
I'm partial to solving with generating functions. Let
[tex]T(x)=\displaystyle\sum_{n\ge0}t_nx^n[/tex]
Multiply both sides of the recurrence by [tex]x^{n+2}[/tex] and sum over all [tex]n\ge0[/tex].
[tex]\displaystyle\sum_{n\ge0}2t_{n+2}x^{n+2}=\sum_{n\ge0}3t_{n+1}x^{n+2}+\sum_{n\ge0}2t_nx^{n+2}[/tex]
Shift the indices and factor out powers of [tex]x[/tex] as needed so that each series starts at the same index and power of [tex]x[/tex].
[tex]\displaystyle2\sum_{n\ge2}2t_nx^n=3x\sum_{n\ge1}t_nx^n+2x^2\sum_{n\ge0}t_nx^n[/tex]
Now we can write each series in terms of the generating function [tex]T(x)[/tex]. Pull out the first few terms so that each series starts at the same index [tex]n=0[/tex].
[tex]2(T(x)-t_0-t_1x)=3x(T(x)-t_0)+2x^2T(x)[/tex]
Solve for [tex]T(x)[/tex]:
[tex]T(x)=\dfrac{2-3x}{2-3x-2x^2}=\dfrac{2-3x}{(2+x)(1-2x)}[/tex]
Splitting into partial fractions gives
[tex]T(x)=\dfrac85\dfrac1{2+x}+\dfrac15\dfrac1{1-2x}[/tex]
which we can write as geometric series,
[tex]T(x)=\displaystyle\frac8{10}\sum_{n\ge0}\left(-\frac x2\right)^n+\frac15\sum_{n\ge0}(2x)^n[/tex]
[tex]T(x)=\displaystyle\sum_{n\ge0}\left(\frac45\left(-\frac12\right)^n+\frac{2^n}5\right)x^n[/tex]
which tells us
[tex]\boxed{t_n=\dfrac45\left(-\dfrac12\right)^n+\dfrac{2^n}5}[/tex]
# # #
Just to illustrate another method you could consider, you can write the second recurrence in matrix form as
[tex]49y_{n+2}=-16y_n\implies y_{n+2}=-\dfrac{16}{49}y_n\implies\begin{bmatrix}y_{n+2}\\y_{n+1}\end{bmatrix}=\begin{bmatrix}0&-\frac{16}{49}\\1&0\end{bmatrix}\begin{bmatrix}y_{n+1}\\y_n\end{bmatrix}[/tex]
By substitution, you can show that
[tex]\begin{bmatrix}y_{n+2}\\y_{n+1}\end{bmatrix}=\begin{bmatrix}0&-\frac{16}{49}\\1&0\end{bmatrix}^{n+1}\begin{bmatrix}y_1\\y_0\end{bmatrix}[/tex]
or
[tex]\begin{bmatrix}y_n\\y_{n-1}\end{bmatrix}=\begin{bmatrix}0&-\frac{16}{49}\\1&0\end{bmatrix}^{n-1}\begin{bmatrix}y_1\\y_0\end{bmatrix}[/tex]
Then solving the recurrence is a matter of diagonalizing the coefficient matrix, raising to the power of [tex]n-1[/tex], then multiplying by the column vector containing the initial values. The solution itself would be the entry in the first row of the resulting matrix.
Show that the equation is exact and find an implicit solution. y cos(xy) + 3x^2 + [x cos(xy) + 2y]y' = 0
We have
[tex]\dfrac{\partial(y\cos(xy)+3x^2)}{\partial y}=\cos(xy)-xy\sin(xy)[/tex]
[tex]\dfrac{\partial(x\cos(xy)+2y)}{\partial x}=\cos(xy)-xy\sin(xy)[/tex]
so the ODE is indeed exact. Then there's a solution of the form [tex]f(x,y)=C[/tex] such that
[tex]\dfrac{\partial f}{\partial x}=y\cos(xy)+3x^2[/tex]
[tex]\implies f(x,y)=\sin(xy)+x^3+g(y)[/tex]
Differentiating wrt [tex]y[/tex] gives
[tex]\dfrac{\partial f}{\partial y}=x\cos(xy)+2y=x\cos(xy)+g'(y)[/tex]
[tex]\implies g'(y)=2y\implies g(y)=y^2+C[/tex]
Then the solution to the ODE is
[tex]f(x,y)=\boxed{\sin(xy)+x^3+y^2=C}[/tex]
You are going to buy a new car worth $25,800. The dealer computes your monthly payment to be $509.55 for 60 months of financing. What is the dealer's effective rate of return on this loan transaction? The dealer's effective rate of return is 1 1%. (Round to one decimal place.)
Answer:
6.9%
Step-by-step explanation:
Interest rate is the one variable in an amortization formula that cannot be determined explicitly. An iterative solution is required, which means the computation must be done by a calculator, spreadsheet, or web site.
My TI-84 TVM Solver tells me that for the given loan amount and payment schedule, the APR is about 6.9%.