Adult and student tickets were sold for a school concert. The adult tickets cost $12 each, and the student tickets cost $8 each. If a total of 360 tickets were sold for $3,480, how many of each kind of ticket were sold?

Answer:

The possible rational zeros for the function are

±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, ±1/2, ±3/2

Step-by-step explanation:

I believe that there is an error in the function with the exponents, it must be:

[tex]h(t) = 2t^{3} + 23t^{2}+59t+24[/tex]

If this is the function that you need, then we must use the rational zero theorem. It says that if  a polynomial function, written in descending order of the exponents, has integer coefficients, then any rational zero must be of the form ± p/ q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Thus

In this case the constant term is 24 and then

p = ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24

The factor of the leading coefficient is 2, thus

q = ±1, ±2

The possible rational zeros for the function are

±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, ±1/2, ±3/2

Answer:

V = int(π(y/3)^2, 0, 8)

(Definite integration of π(y/3)^2 with lower boundary 0 and upper boundary 8)

Step-by-step explanation:

Set up cartesian axis (x and y) to the system.

Let y axis as the line of the centre of the cone, passing through its vertex and the centre of it's circular base. The x axis could be the 90 degree line to the y axis that passes the vertex. So the origin (0,0) is at the vertex.

I'm this setup , looking it as if we are looking it in 2 dimension, we'll see that there is a signature straight line on the x-y plane, which this line will form the cone as it revolute around the y-axis

Find the equation of the line:

Using height 12 and radius base 4, we can get the slope of the line

m = 12/4 = 3

It passes through origin, so the y-intercept is 0

Hence, y = 3x

Since the volume revolves around y-axis, we use the equation volume of revolution around y-axis

V = int(πx^2,a,b)

(Definite integration of πx^2 with lower boundary a and upper boundary b

Since y=3x

x = y/3

For this que

V = int(π(y/3)^2, 0, 8)

(Definite integration of π(y/3)^2 with lower boundary 0 and upper boundary 8)

Answer:

Step-by-step explanation:

The top of the computer desk is rectangular in shape.

Let y represent the width of the rectangle.

The length of the top of the computer desk is 2 1/4 feet longer than its width. Converting 2 1/4 feet to improper fraction, it becomes 9/4 feet. Therefore, the algebraic expression of the length of the of the top of the computer desk in terms of y would be

Length = y + 9/4

Answer:

John- 375

Karen- 125

Step-by-step explanation:

Answer 1...

j =3k

2j + k = 875

substituting the first eqn into the 2nd

2(3k) + k = 875

6k+ k =875

7k = 875

k =875/7 =125

thus j = 3(125) =375

Answer:john weighs 375 pounds.

Karen weighs 125 pounds

Step-by-step explanation:

Let x represent the weight of John.

Let y represent the weight of Karen.

John weighs three times as much as Karen. This means that

x = 3y

Two times John's weight plus Karen's weight is 875 pounds. This means that

2x + y = 875 - - - - - - - -1

Substituting x = 3y into equation 1, it becomes

2 × 3y + y = 875

6y + y = 875

7y = 875

y = 875/7 = 125

Substituting y = 125 into x = 3y. It becomes

x = 3 × 125 = 375

Answer:

299.3

Step-by-step explanation:

Answer:

m-4=26

Step-by-step explanation: I guess and feel like this is correct for some reason

Answer:

P(1,2)

Step-by-step explanation:

There are 2 points.

A(-2,4) and B(7,6)

the point P on the y=2 can also represented as P(x,2)

We can use the distance formula to find the distances AP and BP

[tex]\text{dist} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]

for AP: A(-2,4) and P(x,2)

[tex]AP = \sqrt{(-2 - x)^2 + (4 - 2)^2}[/tex]

[tex]AP = \sqrt{(-2 - x)^2 + 4}[/tex]

[tex]AP = \sqrt{(-1)^2(2 + x)^2 + 4}[/tex]

[tex]AP = \sqrt{(2 + x)^2 + 4}[/tex]

for BP: B(7,6) and P(x,2)

[tex]BP = \sqrt{(7 - x)^2 + (6 - 2)^2}[/tex]

[tex]BP = \sqrt{(7 - x)^2 + 16}[/tex]

the total distance AP + BP will be

[tex]\sqrt{(2 + x)^2 + 4}+\sqrt{(7 - x)^2 + 16}[/tex] (plot is given below)

Our task is to find the value of x such that the above expression is small as possible. (we can find this either through plotting or differentiating)

If you plot the above equation, the minimum point of the curve will be clearly visible, and it will be at x = 1. Hence, the point P(1,2) is such that the total distance AP + BP is as small as possible.

The point P that makes the total distance AP + BP smallest on the line y=2 is given by the x-coordinate of the midpoint of A and B because the shortest distance is in a straight line. Therefore, the point P is (2.5, 2).

To find the point P on the line y = 2 that makes the total distance AP + BP the smallest, you need to recall that the shortest distance between two points is a straight line. So, ideally, we want to find a point P (x,2) that is on the same vertical line (or x-coordinate) that intersects the line AB at the midpoint.

Step 1: Find the midpoint of A and B. The midpoint M is obtained by averaging the x and y coordinates of A and B: M = ((-2+7)/2 , (4+6)/2) = (2.5, 5).

Step 2: Since line y = 2 is horizontal, the x-coordinate of our point P will stay the same with the midpoint x-coordinate. Therefore, P has coordinates (2.5, 2).

So, the point on the line y = 2 that makes the total distance AP + BP as small as possible is P (2.5, 2).

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Answer:

(C) 9900

Step-by-step explanation:

The right triangle which right angle is at P and PR is parallel to the x-axis can have 4 sets (A,B,C,D as illustrated) of format depends on which direction is the right angle located.

each set have

(1+2+3+4+5+6+7+8+9) x (1+2+3+4+5+6+7+8+9+10) = 45 x 55 = 2475 right triangles

4 sets: 2475 x 4 = 9900

Answer:

70/5

Step-by-step explanation:

70/3 and then try 70/4 and then 70/5

The total rent for the week was $840, based on the given conditions of per person cost and the price decrease with additional participants.

The subject of this question is Mathematics, and this is a problem ideally pitched at high school level. It involves constructing equations from the given information to solve the problem.

Let's begin by determining the amount of people who went on the trip initially. We'll call them 'n'. The cost per person on the trip was $70, so that the total cost of the trip is $70n.

Now, if they had persuaded three more people to go (n + 3), the cost per person would've dropped by $14 to $56 which, multiplied by the new total of attendees would still be equal to the total cost of the trip ($56(n + 3)).

As such, we build the equation $70n = $56(n + 3). Here's the breakdown and solving of the equation: $70n = $56n + $168.

Subtracting $56n from both sides gives $14n = $168. Dividing both sides by 14 finally gives n = 12.

To find the total cost of the rent, substitute n = 12 into the equation $70n, yielding $70(12) = $840.

So, the rent for the week was $840.

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Answer:

24

Step-by-step explanation:

3.49 to the nearest whole number is 3

7.508 to the nearest whole number is 8

Multiplying both will yield 3 * 8 = 24

Although it might be mistaken that 3.49 could be approximated to 4, this is absolutely wrong. This is because we round up all values 5 and above after the decimal to 1 while we round down all values less than 5 after the decimal to 0.

Hence be it 3.4999, since it is less than 5, it is rounded as 3 to the nearest whole number digit

Answer:

Step-by-step explanation:

To determine how much are you making on your investment, we will apply the simple interest formula. It is expressed as

I = PRT/100

Where

P = principal or amount invested,

R = interest rate

T = time

From the information given,

P = $1,500

T = 1 year

R = 8.5

I =( 1500×8.5×1)/100

I = $127.5

For a year you owe $1,600 on a credit card. You pay 19% interest a year on this credit card debt.

Interest paid = (1600×19×1)/100 = $304

The loss is 304 - 127.5 = - $176.5

Final answer:

You are making $127.50 on your investment in a year and paying $304 in interest on your credit card debt in a year, resulting in a total loss of $176.50.

Explanation:

First, let's calculate how much money you are making on your investment in a year. To do this, multiply your initial investment of $1,500 by the interest rate of 8.5%: $1,500 * 8.5% = $127.50. So, you are making $127.50 on your investment in a year.

Next, let's calculate how much money you are paying in interest on your credit card debt in a year. To do this, multiply your credit card debt of $1,600 by the interest rate of 19%: $1,600 * 19% = $304. So, you are paying $304 in interest on your card in a year.

To calculate your total gain/loss for the year, subtract the amount you are paying in interest from the amount you are making on your investment: $127.50 - $304 = -$176.50. Therefore, you have a total loss of $176.50 for the year. Remember to enter a negative value for a loss.

Answer:

6:5

Step-by-step explanation:

It is given that a pot worth $2.35 and there are 6 quarters, 5 dimes, 5 pennies, the rest of the coins are nickels.

We know that

$1 = 100 cents

1 penny = 1 cent = $0.01

1 nickel = 5 cents. = $0.05

1 dime = 10 cents. = $0.10

1 quarter = 25 cents = $0.25

The value of 6 quarters is

[tex]6\times 0.25=1.50[/tex]

The value of 5 dimes is

[tex]5\times 0.10=0.50[/tex]

The value of 5 pennies is

[tex]5\times 0.01=0.05[/tex]

Let x be the number of nickels. So, the value of x nickels is

[tex]x\times 0.05=0.05x[/tex]

Total value of 6 quarters, 5 dimes, 5 pennies, and x nickels is

[tex]Total =1.50+0.50+0.05+0.05x[/tex]

[tex]Total =2.05+0.05x[/tex]

It is given that the pot worth is $2.35.

[tex]2.05+0.05x=2.35[/tex]

Subtract 2.05 from both sides.

[tex]0.05x=0.30[/tex]

Divide both sides by 0.05.

[tex]x=6[/tex]

The number of nickels is 5.

[tex]\dfrac{Nickel}{Dimes}=\dfrac{6}{5}=6:5[/tex]

Therefore, the ratio of nickels to dimes is 6:5.

In a pot worth $2.35 containing 6 quarters, 5 dimes, 5 pennies, and some nickels, the ratio of nickels to dimes is 6:5.

To find the ratio of nickels to dimes, we need to determine the number of nickels and dimes in the pot. We know that there are 6 quarters, 5 dimes, and 5 pennies in the pot, which is a total of 16 coins. Therefore, the number of nickels should be the difference between the total number of coins and the sum of quarters, dimes, and pennies.

The total value of the coins in the pot is $2.35. Since 6 quarters are worth $1.50, 5 dimes are worth $0.50, and 5 pennies are worth $0.05, the remaining value should come from the nickels.

Thus, the value of the nickels is $2.35 - $1.50 - $0.50 - $0.05 = $0.30. Since each nickel is worth $0.05, the number of nickels is $0.30 ÷ $0.05 = 6.

The ratio of nickels to dimes is therefore 6:5, which means that for every 6 nickels, there are 5 dimes.

Learn more about ratios here:

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Answer:

10.4

Step-by-step explanation:

8+8*0.3

= 10.4

Hope that helps

Answer:

10.4

Step-by-step explanation:

8+8*0.3

= 10.4

Hope that helps

Answer:

[tex]P_o = \frac{143000}{e^{-20*0.01303024661}}=110193.69[/tex]

And we can round this to the nearest up integer and we got 110194.  

Step-by-step explanation:

The natural growth and decay model is given by:

[tex]\frac{dP}{dt}=kP[/tex]   (1)

Where P represent the population and t the time in years since 1970.

If we integrate both sides from equation (1) we got:

[tex] \int \frac{dP}{P} =\int kdt [/tex]

[tex]ln|P| =kt +c[/tex]

And if we apply exponentials on both sides we got:

[tex]P= e^{kt} e^k [/tex]

And we can assume [tex]e^k = P_o[/tex]

And we have this model:

[tex]P(t) = P_o e^{kt}[/tex]

And for this case we want to find [tex]P_o[/tex]

By 1990 we have t=20 years since 1970 and we have this equation:

[tex]143000 = P_o e^{20k}[/tex]

And we can solve for [tex]P_o[/tex] like this:

[tex]P_o = \frac{143000}{e^{20k}}[/tex]   (1)

By 2019 we have 49 years since 1970 the equation is given by:

[tex]98000 = P_o e^{49k}[/tex]   (2)

And replacing [tex]P_o[/tex] from equation (1) we got:

[tex]98000=\frac{143000}{e^{20k}} e^{49k} =143000 e^{29k}[/tex]  

We can divide both sides by 143000 we got:

[tex]\frac{98000}{143000} =0.685 = e^{29k}[/tex]

And if we apply ln on both sides we got:

[tex]ln(0.685) = 29k[/tex]

And then k =-0.01303024661[/tex]

And replacing into equation (1) we got:

[tex]P_o = \frac{143000}{e^{-20*0.01303024661}}=110193.69[/tex]

And we can round this to the nearest up integer and we got 110194.  

Answer:

Water level after 24 hours will be [tex]\frac{3}{25}\ inches[/tex].

Step-by-step explanation:

Given:

Evaporate rate of water= 0.005 inches per hour.

We need to find the level of water after 24 hours.

now we know the in 1 hour the water evaporates from a pool at a rate of 0.005 inches.

So To find the water level after 24 hrs we will multiply the evaporation rate with total number of hours which is 24 and the divide by 1 hour we get.

Framing in equation form we get;

water level after 24 hrs = [tex]\frac{24 \times 0.005}1 = 0.12\ inches[/tex]

Now to convert the number in rational form we have to multiply and divide the number by 100 we get;

[tex]\frac{0.12\times 100}{100} = \frac{12}{100}=\frac{3}{25}\ inches[/tex]

Hence Water level after 24 hours will be [tex]\frac{3}{25}\ inches[/tex].

Answer: The answer is 6

Step-by-step explanation: this is a combination because it is without repetition.

3!/(3-2)!

(3x2x1)/1!

6/1 = 6

So the answer is 6 different ways

15 % of usual price is a savings

Solution:

Given that tool set normally cost $80

During the sale this week the tools that cost $12 less than usual

To find: what percentage of the usual price is a savings

From given information,

Usual price of tool set = $ 80

Given that tools that cost $12 less than usual which means she saved $ 12

Savings = $ 12

To find what percentage of the usual price is a savings, we can solve by framing a expression,

Let "x" be the required percentage

Then x % of percentage is equal to savings price

x % of usual price = savings price

x % of 80 = 12

[tex]\frac{x}{100} \times 80 = 12\\\\x = \frac{12 \times 100}{80}\\\\x = 15 \%[/tex]

Therefore 15 % of usual price is a savings

Answer:

[tex]6x^2-5x+5[/tex] should be subtracted.

Step-by-step explanation:

we find the polynomial that is subtracted from 7x^2-6x+5 to get the difference equal to x^2-x

7x^2-6x+5-polynomial=x^2-x

To get the polynomial subtract x^2-x from the given polynomial

[tex]7x^2-6x+5 - (x^2-x)=  7x^2-6x+5-x^2+x[/tex]

[tex]6x^2-5x+5[/tex]

So [tex]6x^2-5x+5[/tex] should be subtracted.

Answer:

The answer to your question is letter D

Step-by-step explanation:

Data

slope = m = 2

Point (-3, 4)

Process

1.- Substitute the data in the line equation

                     y - y1 = m(x - x1)

                     y - 4 = 2 (x + 3)

2.- Expand

                     y - 4 = 2x + 6

3.- Solve for y and simplify

                     y = 2x + 6 + 4

                     y = 2x + 10

Answer:

y=2x+10

Step-by-step explanation:

Answer:the unit rate is $0.5

Step-by-step explanation:

Andre rents a bike for a day. The rental changes can be determined by the equation: R = $0.5d, where R is rental changes and d is number of days she rents the bike.

The unit rate multiplied by the number of days for which the bike was rented would give the total rental charge. From the given equation, the unit rate would be $0.5

Answer:

10

Step-by-step explanation:

divide the last number by the previous number

Answer:

10

Step-by-step explanation:

If you calculate it correctly, every number in front of that number it 10 above.

Answer:

Part 1) [tex]x=6[/tex]

Part 2) [tex]x=5,75[/tex]

Part 3) [tex]NO=80\ units[/tex]

Part 4) [tex]x=3,5[/tex]

Step-by-step explanation:

Part 1) Find the value of x

we know that

In a parallelogram opposites sides are congruent and parallel

In this problem

GH=FE

substitute the given values

[tex]2x+10=22[/tex]

solve for x

subtract 10 both sides

[tex]2x=22-10[/tex]

[tex]2x=12[/tex]

Divide by 2 both sides

[tex]x=6[/tex]

Part 2) Find the value of x

we know that

In a parallelogram opposites sides are congruent and parallel

In this problem

FG=EH

substitute the given values

[tex]4x+5=28[/tex]

solve for x

subtract 5 both sides

[tex]4x=28-5[/tex]

[tex]4x=23[/tex]

divide by 4 both sides

[tex]x=5,75[/tex]

Part 3) What is the length of NO?

step 1

Find the value of x

we know that

In a parallelogram opposites sides are congruent and parallel

In this problem

NO=ML

substitute the given values

[tex]4x+20=2x+50[/tex]

solve for x

Group terms

[tex]4x-2x=50-20[/tex]

[tex]2x=30[/tex]

Divide by 2 both sides

[tex]x=15[/tex]

step 2

Find the value of NO

we have that

[tex]NO=4x+20[/tex]

substitute the value of x

[tex]NO=4(15)+20=80\ units[/tex]

Part 4) we know that

The diagonals in a parallelogram bisect each other

so

LB=BN

LN=LB+BN ----> by addition length postulate

LN=2LB

substitute the given values

[tex]2x+5=2(6)[/tex]

solve for x

[tex]2x+5=12[/tex]

subtract 5 both sides

[tex]2x=12-5[/tex]

[tex]2x=7[/tex]

Divide by 2 both sides

[tex]x=3,5[/tex]

Answer:

Option B -  father's age = 52; daughter's age = 13

Step-by-step explanation:

Given : A girl is now one-fourth as old as her father, and in seven years, she will be one-half as old as her father was twelve years ago.

To find : What are her and her father's present ages?

Solution :

Let the father's present age is 'x'.

A girl is now one-fourth as old as her father.

i.e. Girl age is [tex]\frac{x}{4}[/tex]

In seven years, she will be one-half as old as her father was twelve years ago.

i.e. [tex]\frac{x}{4}+7=\frac{1}{2}(x-12)[/tex]

[tex]\frac{x}{4}+7=\frac{x}{2}-6[/tex]

[tex]\frac{x}{4}-\frac{x}{2}=-6-7[/tex]

[tex]\frac{x-2x}{4}=-13[/tex]

[tex]-x=-52[/tex]

[tex]x=52[/tex]

The father's age is 52 years.

The daughter's age is [tex]\frac{52}{4}=13[/tex]

Therefore, option B is correct.

On the first day, a total of 40 items were sold for $356. Pies cost $10 and cakes cost $8. Define the variables, write a system of equations to find the number of cakes and pies sold, and state how many pies were sold.

The variables are defined as:

"c" represent the number of cakes sold and "p" represent the number of pies sold

The system of equations used are:

c + p = 40 and 8c + 10p = 356

18 pies and 22 cakes were sold

Let "c" represent the number of cakes sold

Let "p" represent the number of pies sold

Cost of 1 pie = $ 10

Cost of 1 cake = $ 8

Given that total of 40 items were sold

number of cakes + number of pies = 40

c + p = 40 ------ eqn 1

Given items were sold for $356

number of cakes sold x Cost of 1 cake + number of pies sold x Cost of 1 cake = 356

[tex]c \times 8 + p \times 10 = 356[/tex]

8c + 10p = 356  ----- eqn 2

Let us solve eqn 1 and eqn 2

From eqn 1,

p = 40 - c    ---- eqn 3

Substitute eqn 3 in eqn 2

8c + 10(40 - c) = 356

8c + 400 - 10c = 356

-2c = - 44

c = 22

Substitute c = 22 in eqn 3

p = 40 - c

p = 40 - 22

p = 18

Thus 18 pies and 22 cakes were sold

Answer:

[tex]a_n = 2^{\frac{n^2-1}{4} + 1} + \frac{2^{n^2} - \, 2^{\frac{n^2-1}{4} + 1}}{4}[/tex]

For n = 3, there are 134 possibilities

Step-by-step explanation:

First, lets calculate the generating function.

For each square we have 2 possibilities: red and blue. The Possibilities between n² squares multiply one with each other, giving you a total of 2^n² possibilities to fill the chessboard with the colors blue or red.

However, rotations are to be considered, then we should divide the result by 4, because there are 4 ways to flip the chessboard (including not moving it), that means that each configuration is equivalent to three other ones, so we are counting each configuration 4 times, with the exception of configurations that doesnt change with rotations.

A chessboard that doesnt change with rotations should have, in each position different from the center, the same colors than the other three positions it could be rotated into. As a result, we can define a symmetric by rotations chessboard with only (n²-1)/4 + 1 squares (the quarter part of the total of squares excluding the center plus the center).

We cocnlude that the total of configurations of symmetrical boards is [tex] 2^{\frac{n^2-1}{4} + 1} [/tex]

Since we have to divide by 4 the rest of configurations (because we are counted 4 times each one considering rotations), then the total number of configutations is

[tex]a_n = 2^{\frac{n^2-1}{4} + 1} + \frac{2^{n^2} - \, 2^{\frac{n^2-1}{4} + 1}}{4}[/tex]

If n = 3, then the total amount of possibilities are

[tex]a_3 = 2^{\frac{3^2-1}{4} + 1} + \frac{2^{3^2} - \, 2^{\frac{3^2-1}{4} + 1}}{4} =  134[/tex]

Answer:

  8

Step-by-step explanation:

Fill in the variable value and do the arithmetic.

  [tex]\dfrac{2^5}{2^2}=\dfrac{32}{4}=8[/tex]

___

Of course, the fraction can be simplified first:

  [tex]\dfrac{x^5}{x^2}=x^{5-2}=x^3\\\\2^3=8[/tex]

To evaluate the expression, substitute x with 2, simplify the exponents, and perform the multiplication

To evaluate the expression \dfrac{x^5}{x^2} \times 2 \times 5

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Answer:

Part 1) [tex]\$106[/tex]

Part 2) [tex]\$53[/tex]

Part 3) [tex]\$212[/tex]

Part 4) [tex]\$132.50[/tex]

Part 5) [tex]\$1.06x[/tex]

Step-by-step explanation:

we have

Money in a particular savings account increases by about 6% after a year.

we know that

The simple interest formula is equal to

[tex]A=P(1+rt)[/tex]

where

A is the Final Investment Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

Part 1) How much money will be in the account after one year if the initial amount is $100

in this problem we have

[tex]t=1\ year\\ P=\$100\\ A=?\\r=6\%=6/100=0.06[/tex]

substitute in the formula above

[tex]A=100(1+0.06*1)[/tex]

[tex]A=100(1.06)[/tex]

[tex]A=\$106[/tex]

Part 2) How much money will be in the account after one year if the initial amount is $50

in this problem we have

[tex]t=1\ year\\ P=\$50\\ A=?\\r=6\%=6/100=0.06[/tex]

substitute in the formula above

[tex]A=50(1+0.06*1)[/tex]

[tex]A=50(1.06)[/tex]

[tex]A=\$53[/tex]

Part 3) How much money will be in the account after one year if the initial amount is $200

in this problem we have

[tex]t=1\ year\\ P=\$200\\ A=?\\r=6\%=6/100=0.06[/tex]

substitute in the formula above

[tex]A=200(1+0.06*1)[/tex]

[tex]A=200(1.06)[/tex]

[tex]A=\$212[/tex]

Part 4) How much money will be in the account after one year if the initial amount is $125

in this problem we have

[tex]t=1\ year\\ P=\$125\\ A=?\\r=6\%=6/100=0.06[/tex]

substitute in the formula above

[tex]A=125(1+0.06*1)[/tex]

[tex]A=125(1.06)[/tex]

[tex]A=\$132.50[/tex]

Part 5) How much money will be in the account after one year if the initial amount is $x

in this problem we have

[tex]t=1\ year\\ P=\$x\\ A=?\\r=6\%=6/100=0.06[/tex]

substitute in the formula above

[tex]A=x(1+0.06*1)[/tex]

[tex]A=x(1.06)[/tex]

[tex]A=\$1.06x[/tex]

Answer:

[tex]1680[/tex]

Step-by-step explanation:

we need a 4-digit number from the numbers [tex]2,3,4,5,6,7,8\ and\ 9[/tex] (without repetition ).

possible number at thousand place [tex]=8[/tex]

Possible numbers at hundred place[tex]=8-1=7[/tex]

Possible numbers at [tex]10^{th}[/tex] place [tex]=7-1=6[/tex]

possible number at unit place [tex]=6-1=5[/tex]

So total possible numbers

[tex]=8\times7\times6\times5\\=1680[/tex]

Other method :

We are taking [tex]4[/tex] numbers out of [tex]8[/tex] and here order matters so we will use permutation.

Total possible numbers [tex]=^8P_{4}[/tex]

[tex]\frac{8!}{(8-4)!}\\ =\frac{8!}{4!}\\ =8\times7\times6\times5\\=1680[/tex]

Answer:

Part a) The graph in the attached figure (see the explanation)

Part b) The friend is not correct

Step-by-step explanation:

The questions are

a. Write and graph a system of linear equations that  represent the  amounts in each of your savings  accounts

b. Your friend says that in 10 weeks you will both have the same amount of money in your savings  accounts. Is your friend correct? Use the graph from part (a) to explain your answer.

Part a)

Let

x ----> the number of weeks

y ---> the amount in the saving account

we know that

The linear equation in slope intercept form is equal to

[tex]y=mx+b[/tex]

where

m is the slope or unit rate of the linear equation

b is the y-intercept or initial value of the linear equation

In this problem we have

Your saving accounts

The slope is equal to [tex]m=\$10\ each\ week[/tex]

The y-intercept is equal to [tex]b=\$90[/tex]

substitute

[tex]y=10x+90[/tex] ----> equation A

Your friend saving accounts

The slope is equal to [tex]m=\$15\ each\ week[/tex]

The y-intercept is equal to [tex]b=\$35[/tex]

substitute

[tex]y=15x+35[/tex] ----> equation B

using a graphing tool

the graph in the attached figure

Part b) Your friend says that in 10 weeks you will both  have the same amount of money in your savings  accounts. Is your friend correct? Use the graph  from part (a) to explain your answer

we know that

When solving a system of equations by graphing, the solution of the system is the intersection point both graphs

In this problem, the intersection point is (11,200)

That means ----> In 11 weeks, both you and your friend have the same amount of money saved up, $200

Therefore

The friend is not correct

Answer:

Part 1) The length of two sides and the measure of the included angle (Side-Angle-Side)

Part 2) [tex]b^2=a^2+c^2-2(a)(c)cos(B)[/tex]  

Part 3) [tex]b=11.6\ in[/tex]

Step-by-step explanation:

we have

In the triangle ABC

[tex]a=11\ in\\c=9\ in\\B=70^o[/tex]

Part 1) Which information about the triangle is given?

In this problem we have the length of two sides and the measure of the included angle (Side-Angle-Side)

see the attached figure to better understand the problem

Part 2) Which formula can you use ti find b?

I can use the law of cosines

[tex]b^2=a^2+c^2-2(a)(c)cos(B)[/tex]  

we have

[tex]a=11\ in\\c=9\ in\\B=70^o[/tex]

substitute the given values

[tex]b^2=11^2+9^2-2(11)(9)cos(70^o)[/tex]

[tex]b^2=202-67.72[/tex]  

[tex]b=11.59\ in[/tex]

Part 3) What is b, rounded to the nearest tenth?

Remember that

To Round a number

a) Decide which is the last digit to keep  

b) Leave it the same if the next digit is less than [tex]5[/tex] (this is called rounding down)  

c) But increase it by  [tex]1[/tex] if the next digit is  [tex]5[/tex] or more (this is called rounding up)

In this problem we have

[tex]11.59\ in[/tex]

We want to keep the digit [tex]5[/tex]

The next digit is [tex]9[/tex] which is 5 or more, so increase the "5" by 1 to "6"

therefore

[tex]b=11.6\ in[/tex]