Solving Quadratic Equations by completing the square:

z^2 - 3z - 5 = 0

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]

Answer:

5 pounds of nuts and 20 pounds of pretzels

Step-by-step explanation:

Let

x ----> the number of pounds of nuts

y ----> the number of pounds of pretzels

we know that

x+y=25

x=25-y ------> equation A

4x+2y=60 ---> equation B

Solve the system by substitution

Substitute equation A in equation B and solve for y

4(25-y)+2y=60

100-4y+2y=60

4y-2y=100-60

2y=40

y=20 pounds of pretzels

Find the value of x

x=25-y

x=25-20=5 pounds of nuts

Final answer:

Dave should use 5 pounds of nuts and 20 pounds of pretzels to make the party mix.

Explanation:

Let's assume that Dave buys x pounds of nuts and y pounds of pretzels. Since he wants to buy a total of 25 pounds of party mix, we can write the equation x + y = 25.

The cost of nuts per pound is $4 and the cost of pretzels per pound is $2. Therefore, the cost of x pounds of nuts is 4x dollars and the cost of y pounds of pretzels is 2y dollars.

We can write the equation 4x + 2y = 60 to represent the total cost of the party mix.

To solve this system of equations, we can use substitution. Solve the first equation for x in terms of y: x = 25 - y.

Substitute this expression for x into the second equation to get 4(25 - y) + 2y = 60.

Simplify this equation to get 100 - 4y + 2y = 60. Combine like terms to get -2y = -40. Divide both sides by -2 to solve for y: y = 20.

Substitute this value back into the first equation to find x: x = 25 - 20 = 5.

Therefore, Dave should use 5 pounds of nuts and 20 pounds of pretzels to make the party mix.

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.

Answer:

The Annual percentage yield is 3.25%.

Step-by-step explanation:

Given : Money invested at the rate of 3.2% compounded continuously.

To find : What's the annual percentage yield?

Solution :

Money invested at the rate of 3.2% compounded continuously.

The compounded continuously formula is

[tex]A=Pe^{rt}[/tex]

Where, P is the principal P=1

t is the time t=1

r is the interest rate r=3.2%=0.032

Substitute the value in the formula,

[tex]A=Pe^{rt}[/tex]

[tex]A=1\times e^{0.032}[/tex]

[tex]A=1.0325[/tex]

The Annual percentage yield is

[tex]APY=(A-1)\times 100[/tex]

[tex]APY=(1.0325-1)\times 100[/tex]

[tex]APY=0.0325\times 100[/tex]

[tex]APY=3.25\%[/tex]

Therefore, The Annual percentage yield is 3.25%.

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

Answer:

Using the critical value rule, if a two-sided null hypothesis is rejected for a single mean at a given significance level, the corresponding one-sided null hypothesis will "always" be rejected at the same significance level.

Step-by-step explanation:

Consider the provided statement.

As the value of p is less than the significance level, therefore always reject the null hypothesis. Where p is exact level of significance.

Therefore, the answer to the statement is "Using the critical value rule, if a two-sided null hypothesis is rejected for a single mean at a given significance level, the corresponding one-sided null hypothesis (i.e., the same sample size, the same standard deviation, and the same mean) will always be rejected at the same significance level."

If a two-sided null hypothesis is rejected at a given significance level, the corresponding one-sided null hypothesis will also be rejected at the same significance level, because the two-sided test is more stringent.

Using the critical value rule, if a two-sided null hypothesis is rejected for a single mean at a given significance level, the corresponding one-sided null hypothesis will also be rejected at the same significance level.

This is because when performing a two-sided test, we are testing both ends of the distribution, thus it requires a stricter criteria to reject the null hypothesis than a one-sided test. Since we have already rejected it under a stricter evaluation, we will definitely reject it under a less strict one.

Consider an example where you are using a significance level of 5 percent (α = 0.05). Suppose that your computed t-statistic is 2.2. This value is greater than the critical value for a two-tailed test from the t29 distribution, which is 2.045. Therefore, you reject the two-sided null hypothesis.

Consequently, when comparing your t-statistic (2.2) with the critical value for a one-sided test (which will be less stringent than that for a two-tailed test), you also reject the one-sided null hypothesis.

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Answer:

$2000 a year.

Step-by-step explanation:

Let's find the answer by using the following formula:

taxes=(house assessment)*(tax rate) for the initial conditions we have:

(1600/year)=(64000)*(tax rate)

(1600/year)/(64000)=(tax rate)

tax rate=0.025/year

For the current conditions we have:

taxes=(house assessment)*(tax rate)

taxes=(80000)*(0.025/year)

taxes=2000/year

So, the taxes will be $2000 a year.

Answer:

The results do not contradict expectations.

Step-by-step explanation:

Given that a genetic experiment with peas resulted in one sample of offspring that consisted of 447 green peas and 172 yellow peas.

Proportion of yellow peas = [tex]\frac{172}{172+447} =27.79%[/tex]

Std error = 0.25(0.75)/sq rt 619

=0.0174

Proportion difference = 0.2779-0.25=0.0279

Test statistic = 0.0279/0.0174 =1.603

p value = 0.1089

For two tailed we have p value >0.10

Hence accept null hypothesis.

The results do not contradict expectations.

Answer with Step-by-step explanation:

Since we have given that

AB = 0 and A is invertible so, AA⁻¹ = I

So, Consider,

[tex]AB=0[/tex]

Multiplying A⁻¹ on both the sides, we get that

[tex]A^{-1}AB=A^{-1}0\\\\(AA^{-1})B=0\\\\IB=0\\\\B=0[/tex]

Hence proved.

Answer: [tex]-3\dfrac{1}{2}[/tex].

Step-by-step explanation:

We know that the slope of a line passing through points (a,b) and (c,d) is given by :_

[tex]m=\dfrac{d-b}{c-a}[/tex]

The given points : (-4,3) and (-2, -4)

Now, the slope of the line passing through points (-4,3) and (-2, -4) is given by :-

[tex]m=\dfrac{-4-3}{-2-(-4)}\\\\\Rightarrow\ m=\dfrac{-7}{-2+4}\\\\\Rightarrow\ m=\dfrac{-7}{2}=-3\dfrac{1}{2}[/tex]

The slope of a line passing through points (-4,3) and (-2, -4) is [tex]-3\dfrac{1}{2}[/tex].

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.

Answer:

  a) uniform

  b) 1/2

  c) 1/1000

Step-by-step explanation:

a) "numbers with equal​ probability" have a uniform distribution.

__

b) Even numbers make up 1/2 of all numbers.

__

c) There are ten such numbers in the range, so the probability is ...

  10/10000 = 1/1000

The selection of telephone numbers can be modeled using a Uniform distribution. The probability of selecting an even number is 1/2, while the chance of selecting a number ending in 666 is 0.001.

The questions asked can be explained using probability theory, a branch of mathematics.

a) To model the selection of the telephone numbers, one would use a Uniform distribution. This is because every number in the range has an equal chance of being selected.

b) The probability that the selected number is even relies on the last digit of the telephone number. As the last digit could be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, each with equal probability, the chance that it is even (0, 2, 4, 6, or 8) is 1/2 or 50%.

c) The probability that the selected number ends in 666 is much lower. Since there are 10,000 possible numbers, and only 10 of them end in 666 (443-0666, 443-1666, etc. through 443-9666), the probability is 10 in 10,000 or 0.001 (0.1%).

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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.

Answer:

At 95% confidence level, she used 11 people to estimate the confidence interval

Step-by-step explanation:

The bounds of the confidence interval are: 740 to 920

Mean is calculated as the average of the lower and upper bounds of the confidence interval. So, for the given interval mean would be:

[tex]u=\frac{740+920}{2}=830[/tex]

Margin of error is calculated as half of the difference between the upper and lower bounds of the confidence interval. So, for given interval, Margin of Error would be:

[tex]E=\frac{920-740}{2}=90[/tex]

Another formula to calculate margin of error is:

[tex]E=z\frac{\sigma}{\sqrt{n}}[/tex]

Standard deviation is given to be 150. Value of z depends on the confidence level. Confidence Level is not mentioned in the question, but for the given scenario 95% level would be sufficient enough.

z value for this confidence level = 1.96

Using the values in above formula, we get:

[tex]90=1.96 \times \frac{150}{\sqrt{n} }\\\\ n = (\frac{1.96 \times 150}{90})^{2}\\\\ n=11[/tex]

So, at 95% confidence level her assistant used a sample of 11 people to determine the interval estimate

Final answer:

The sample size used by the assistant to determine the interval estimate is 7.

Explanation:

To determine how large a sample the assistant used to determine the interval estimate, we need to use the formula for the margin of error:

Margin of Error = Critical Value × Standard Deviation / sqrt(Sample Size)

In this case, the margin of error is half the width of the interval estimate, which is (920 - 740) / 2 = 90.

Using a z-table, the critical value for a 95% confidence level is approximately 1.96.

By substituting the given values into the formula, we can solve for the sample size:

90 = 1.96 × 150 / sqrt(Sample Size)

Simplifying the equation, we get:

sqrt(Sample Size) = 1.96 × 150 / 90

Sample Size = (1.96 × 150 / 90)^2 = 6.83

Since we cannot have a fraction of a sample, we round up to the nearest whole number.

Therefore, the assistant used a sample size of 7 to determine the interval estimate.

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

Answer:

0.0358

Step-by-step explanation:

In a 52 deck, 13 cards are Clubs, and 26 cards are red.

There are ₁₃C₃ ways to choose 3 Clubs from 13.

There are ₂₆C₂ ways to choose 2 red cards from 26.

There are ₅₂C₅ ways to choose 5 cards from 52.

P = (₁₃C₃ ₂₆C₂) / ₅₂C₅

P = (286 × 325) / 2598960

P = 0.0358

Answer:14

Step-by-step explanation:

A nickel is 5 cents (20% of dollar)

and a quarter is 25 cents (25% of dollar)

We have given a pile of 42 coins worth of $4.90

Let x be the no nickels and

y be the no of quarter

therefore

x+y=42    -----1

[tex]\frac{x}{4}[/tex]+[tex]\frac{y}{20}[/tex]=4.90 ---2

Solving [tex]\left ( 1\right )&\left ( 2\right )[/tex] we get

x=14 & y=28

Therefore no of nickels is 14 & no of quarters is 28

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]

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 }

The price of the house = $ 100000

The down payment is 20 % of 100000 means [tex]0.20\times100000=20000[/tex] dollars

So, loan amount will be = [tex]100000-20000=80000[/tex] dollars

Case A:

30-year mortgage at a rate of 7 %

p = 80000

r = [tex]7/12/100=0.005833[/tex]

n = [tex]30\times12=360[/tex]

EMI formula is :

[tex]\frac{p\times r\times(1+r)^n}{(1+r)^n-1}[/tex]

Putting the values in formula we get;

[tex]\frac{80000\times0.005833\times(1+0.005833)^360}{(1+0.005833)^360-1}[/tex]

= [tex]\frac{80000\times0.005833\times(1.005833)^360}{(1.005833)^360-1}[/tex]

Monthly payment = $532.22

So, total amount paid in 30 years will be = [tex]532.22\times360=191599.20[/tex]

Interest paid will be = [tex]191599.20-100000=91599.20[/tex] dollars

Case B:

15-year mortgage at a rate of 7 %.

Here everything will be same as above. Only n will change.

n = [tex]15\times12=180[/tex]

Putting the values in formula we get;

[tex]\frac{80000\times0.005833\times(1+0.005833)^180}{(1+0.005833)^180-1}[/tex]

= [tex]\frac{80000\times0.005833\times(1.005833)^180}{(1.005833)^180-1}[/tex]

Monthly payment = $719.04

Total amount paid in 15 years will be = [tex]719.04\times180=129427.20[/tex]

Interest paid will be = [tex]129427.20-100000=29427.20[/tex] dollars

To find the monthly payment and total amount of interest paid for each mortgage, use the formula A = P(1+r/12)^(12n) / (12n), where A is the monthly payment, P is the principal, r is the interest rate, and n is the number of months.

To find the monthly payment and total amount of interest paid for each mortgage option, we can use the formula for calculating the monthly mortgage payment:

A = P(1+r/12)^(12n) / (12n)

where A is the monthly payment, P is the principal (price of the house minus the down payment), r is the interest rate (expressed as a decimal), and n is the number of months in the mortgage term.

For option a) the 30-year mortgage, we have:

P = 100000 - (0.2 * 100000) = $80000

r = 0.07

n = 30 * 12 = 360

Plugging these values into the formula, we get:

A = (80000(1+(0.07/12))^(12 * 30)) / (12 * 30) = $532.09

To calculate the total amount of interest paid, we subtract the principal from the total payment over the life of the mortgage:

Total Interest Paid = (360 * 532.09) - 80000 = $93891.24

For option b) the 15-year mortgage, we have:

P = 100000 - (0.2 * 100000) = $80000

r = 0.07

n = 15 * 12 = 180

Plugging these values into the formula, we get:

A = (80000(1+(0.07/12))^(12 * 15)) / (12 * 15) = $754.56

To calculate the total amount of interest paid, we subtract the principal from the total payment over the life of the mortgage:

Total Interest Paid = (180 * 754.56) - 80000 = $75822.80

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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.

the answer is 17. the answer is 17

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

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).

Answer:

The vertex (h,k) is (-4,-7).

Step-by-step explanation:

I assume you are looking for the vertex [tex]y=-4(x+4)^2-7[/tex].

The vertex form of a quadratic is [tex]y=a(x-h)^2+k[/tex] where the vertex is (h,k) and a tells us if the parabola is open down (if a<0) or up (if a>0). a also tells us if it is stretched or compressed.

Anyways if you compare [tex]y=-4(x+4)^2-7[/tex] to [tex]y=a(x-h)^2+k[/tex] , you should see that [tex]a=-4,h=-4,k=-7[/tex].

So the vertex (h,k) is (-4,-7).

Answer:

The vertex is [tex](-4,-7)[/tex]

Step-by-step explanation:

The vertex form of a parabola is given by:

[tex]y=a(x-h)^2+k[/tex], where (h,k) is the vertex and [tex]a[/tex] is the leading coefficient.

The given parabola has equation:

[tex]y=-1(x+4)^2-7[/tex]

When we compare to the vertex form, we have

[tex]-h=4\implies h=-4[/tex] and [tex]k=-7[/tex].

Therefore the vertex is (-4,-7)

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.  

Answer:

radius 5

center (0,-2)

Step-by-step explanation:

The goal is to get to [tex](x-h)^2+(y-k)^2=r^2 \text{ where } (h,k) \text{ is the center and } r \text{ is the radius }[/tex].

We will need to complete the square for both parts.

That is we need to use:

[tex]u^2+bu+(\frac{b}{2})^2=(u+\frac{b}{2})^2[/tex].

First step is group the x's and y's together and put the constant on the opposing side.  The x's and y's are already together.  So we need to add 21 on both sides:

[tex]x^2+y^2+4y=21[/tex]

Now the x part is already done.

If you compare y^2+4y to [tex]u^2+bu+(\frac{b}{2})^2=(u+\frac{b}{2})^2[/tex]

on the left side we have b is 4 so we need to add (4/2)^2 on both sides of [tex]x^2+y^2+4y=21[/tex].

[tex]x^2+y^2+4y+(\frac{4}{2})^2=21+(\frac{4}{2})^2[/tex]

Now we can write the y part as something squared still using my completing the square formula:

[tex]x^2+(y+\frac{4}{2})^2=21+2^2[/tex]

[tex]x^2+(y+2)^2=21+4[/tex]

[tex](x-0)^2+(y+2)^2=25[/tex]

The center is (0,-2) and radius is [tex]\sqrt{25}=5[/tex]

The answer is:

Center: (0,-2)

Radius: 2.5 units.

To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

Where,

"h" and "k"are the coordinates of the center of the circle and "r" is its radius.

So, we need to complete the square for both variable "x" and "y".

The given equation is:

[tex]x^2+y^2+4y-21=0[/tex]

So, solving we have:

[tex]x^2+y^2+4y=21[/tex]

[tex]x^2+(y^2+4y+(\frac{4}{2})^{2})=21+(\frac{4}{2})^{2}\\\\x^2+(y^2+4y+4)=21+4\\\\x^2+(y^2+2)=25[/tex]

[tex]x^2+(y^2-(-2))=25[/tex]

Now, we have that:

[tex]h=0\\k=-2\\r=\sqrt{25}=5[/tex]

So,

Center: (0,-2)

Radius: 5 units.

Have a nice day!

Note: I have attached a picture for better understanding.

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

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.

Answer:

  80°

Step-by-step explanation:

The sum of the two angles (red and blue) is 145°, so you have ...

  (4x +5)° +(6x -10)° = 145°

  10x = 150 . . . . . . . . divide by °, add 5, simplify

  x = 15 . . . . . . . . . . . divide by 10

Then the measure of the angle of interest is ...

  m∠XMN = (6x -10)° = (6·15 -10)° = 80°