We need to find the length of each board, so that one piece is five times as long as than the other. Let x represent the length of the longer
piece in feet, and let y represent the length of the shorter piece in feet.
Since the length of the board is 24 ft, we can form the following equation.
The length of the longer piece plus the length of the shorter piece equals 24 feet.
Write this equation using the defined variables for each piece. (Enter simplified answers.)

By the principle of mathematical induction, the statement holds for all positive integers n.

Using mathematical induction:

Base case:

For n=1, results into:

2 = 2^2 + 1 - 2

which is true.

Inductive step:

For some positive integer k, we have:

2+4+8+...+2^k = 2^(k+1) + 1 - 2

This implies the statement for n=k+1, i.e.,

2+4+8+...+2^k+2^(k+1) = 2^(k+2) + 1 - 2

From the left-hand side of the equation, we can rewrite it as:

2+4+8+...+2^k+2^(k+1) = (2+4+8+...+2^k) + 2^(k+1)

Applying the induction hypothesis, substitute the expression for 2+4+8+...+2^k:

2+4+8+...+2^k+2^(k+1) = (2^(k+1) + 1 - 2) + 2^(k+1)

Simplify:

2+4+8+...+2^k+2^(k+1) = 2^(k+2) - 1

Using the formula for the sum of a geometric series to simplify the right-hand side of the original statement:

2^(k+2) + 1 - 2 = 2^(k+2) - 1

Thus, the statement holds for n=k+1, supposing it holds for n=k. By the principle of mathematical induction, the statement holds for all positive integers n.

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If the sample size was increased, the 95% confidence interval would decrease. A larger sample size would provide more precise and accurate data, resulting in a narrower confidence interval.

A confidence interval is a range of values within which a population parameter is estimated to lie with a certain level of confidence. It is commonly used in statistical inference to estimate the true value of a population parameter based on a sample from that population.

If the sample size was increased, the 95% confidence interval would likely decrease. This is because a larger sample size typically leads to more precise estimates and less variability in the data, resulting in a narrower confidence interval. However, the exact size of the decrease would depend on various factors such as the amount of variability in the data and the level of statistical significance chosen for the study.

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Answer:

Step-by-step explanation:
150/2 = 75
500/75 = 6.6666666667 hours

it will take 6.2/3 hours to reach 500 bacteria cells

NOT SURE ABOUT EXPERSION BUT YOU COULD TRY THIS

p = 75t

Given that a culture of bacteria grows at a rate proportional to its size. Where the culture starts with 50 cells, then grows 150 after time, "t" equals 2 hours.

We are asked to:

> (a) Find an expression, P(t), to model the number of cells present after "t" hours.

> (b) Determine the time at which the population is at 500 cells.

For part (a):

Since the culture of bacteria grows at a rate proportional to its size, we can model it as the following differential equation.

[tex]\Rightarrow \frac{dP}{dt}=kP; \ Where \ P =P_0 \ at \ t=0 \ and \ P=3P_0 \ at \ t=2[/tex]

Solve the first-order separable differential equation with the given initial condition.

[tex]\Longrightarrow \frac{dP}{dt}=kP \Longrightarrow \frac{1}{P}dP=kdt \Longrightarrow \int\limits {\frac{1}{P} } \, dP=\int\ {k} \, dt \Longrightarrow ln(P)=kt+c[/tex]

[tex]\Longrightarrow e^{ln(P)}=e^{kt}+e^{c} \Longrightarrow P=ce^{kt}[/tex]

Plug in the initial condition.

[tex]\Longrightarrow P_0=ce^{k(0)} \Longrightarrow P_0=c(1) \Longrightarrow \boxed{c=P_0}[/tex]

[tex]\Longrightarrow P=ce^{kt} \Longrightarrow \boxed{P=P_0e^{kt}}[/tex]

Use the second initial condition to find "k."

[tex]\Longrightarrow 3P_0=P_0e^{k(2)} \Longrightarrow 3=e^{k(2)} \Longrightarrow 3=e^{2k} \Longrightarrow ln(3)=ln(e^{2k})[/tex]

[tex]\Longrightarrow k=\frac{ln(3)}{2} \Longrightarrow \boxed{k \approx 0.5493}[/tex]

Thus, the equation to model the situation is,

[tex]\boxed{\boxed{P(t)=50e^{0.5493t}}} \therefore Sol.[/tex]

For part (b):

[tex]P=10P_0[/tex]

[tex]\Rightarrow 10P_0=P_0e^{0.5493t} \Longrightarrow 10=e^{0.5493t} \Longrightarrow ln(10)=ln(e^{0.5493t})[/tex]

[tex]\Longrightarrow ln(10)=0.5493t \Longrightarrow t=\frac{ln(10)}{0.5493} \Longrightarrow \boxed{t=4.192 \ hrs}[/tex]

Thus, the time it takes the population to reach 500 is approx. 4.192 hours.

The total surface area of the composite figure is: 288.62 mm²

From the attached image, we can see that the composite figure is made up of 2 triangles and one rectangle. Thus:

Formula for area of rectangle is:

A = Length * Width

Formula for area of triangle is:

A = ¹/₂ * base * height

Using Pythagoras theorem, length of rectangle is:

L = √(10.4² + 15.3²)

L = 18.5 mm

Thus:

TSA = (18.5 * 7) + 2(¹/₂ * 15.3 * 10.4)

TSA = 129.5 + 159.12

TSA = 288.62 mm²

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Answer:

8 over 15 multiplied by 7 over 14 is equal to 56 over 210

8 over 15 plus 7 over 14 is equal to 217 over 210

8 over 15 plus 8 over 15 is equal to 16 over 15

8 over 15 multiplied by 8 over 15 is equal to 64 over 225

She's taking it out and then she replaces it

and so that means the denominator would stay 15 both times

and since it's a white chip in both draws, it would be 8/15 * 8/15

Answer:

Starting with the equation:

(x + 1)^2 = 13/4

We can use the square root property, which states that if a^2 = b, then a is equal to the positive or negative square root of b.

Taking the square root of both sides, we get:

x + 1 = ±√(13/4)

Simplifying under the radical:

x + 1 = ±(√13)/2

Now we can solve for x by subtracting 1 from both sides:

x = -1 ± (√13)/2

Therefore, the solutions to the equation are:

x = -1 + (√13)/2 or x = -1 - (√13)/2

Step-by-step explanation:

Based on the information, bIda would owe $3700 at the end of 12 months.

Ida would owe $3808 at the end of 12 months.

The formula will be:

I = P * r * t

I = 2500 * 0.04 * 12 = $1200

So, Ida would owe $2500 + $1200 = $3700 at the end of 12 months.

b. For option b, , the monthly interest rate is:

r = 0.01 * 4.33 = 0.0433

Using the same simple interest expression:

I = 2500 * 0.0433 * 12 = $1308

Ida would owe $2500 + $1308 = $3808 at the end of 12 months.

Therefore, option a is the better choice due to the fact she would owe less in total with a simple interest rate of 4% annually.

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Answer:

90

Step-by-step explanation:

Answer:

It seems like the chat transitioned to a different topic. However, based on the search results, it appears that the query was related to solving distance problems using linear equations. One common application of linear equations is in distance problems, where you can create and solve linear equations to find the distance between two points or the rate of travel. Here's an example problem:

Joe drove from city A to city B, which are 120 miles apart. He drove part of the distance at 60 miles per hour (mph) and the rest at 40 mph. If the entire trip took three hours, how many miles did he drive at each speed?

To solve this problem, you can use a system of two linear equations. Let x be the number of miles driven at 60 mph, and y be the number of miles driven at 40 mph. Then you have:

x + y = 120 (total distance is 120 miles) x/60 + y/40 = 3 (total time is 3 hours)

To solve for x and y, you can multiply the second equation by 120 to eliminate fractions and then use the first equation to solve for one of the variables. For example:

x/60 + 3y/120 = 3 x/60 + y/40 = 3 2x/120 + 3y/120 = 3 x/60 + y/40 = 3 x/60 = 3 - y/40 x = 180 - 3y/2 (from the first equation)

Substitute the expression for x into the second equation and solve for y:

x/60 + y/40 = 3 (180 - 3y/2)/60 + y/40 = 3 3 - 3y/160 + y/40 = 3 3 - 3y/160 = 2.75 -3y/160 = -0.25 y = 20

Substitute y = 20 into the expression for x to get:

x = 180 - 3y/2 x = 120

Therefore, Joe drove 120 - 20 = 100 miles at 60 mph and 20 miles at 40 mph.

Step-by-step explanation:

Step-by-step explanation:

Area of a traingle = 1/2 * base * height

   area = 1/2 * 6 * 4 = 12 cm^2

The equation for g(x) in vertex form is g(x) = (x - 2)^ + 3

From the question, we have the following parameters that can be used in our computation:

The graph of g(x)

Also, we have

The graph of g(x) is a translation of f(x)=x^2

From the graph, we can see that

Using the above as a guide, we have the following:

g(x) = f(x - 2) + 3

Substitute the known values in the above equation, so, we have the following representation

g(x) = (x - 2)^ + 3

Hence, the equation for g(x) in vertex form is g(x) = (x - 2)^ + 3

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The cumulative probability is

P(at most 2 ) = 0.187560896 [answer]

a)

Note that the probability of x successes out of n trials is

P(n, x) = nCx p^x (1 - p)^(n - x)

where

n = number of trials = 12

p = the probability of a success = 0.33

x = the number of successes = 3

Thus, the probability is

P ( 3 ) = 0.21509867 [answer]

***********

b)

Note that P(at least x) = 1 - P(at most x - 1).

Using a cumulative binomial distribution table or technology, matching

n = number of trials = 12

p = the probability of a success = 0.33

x = our critical value of successes = 4

Then the cumulative probability of P(at most x - 1) from a table/technology is

P(at most 3 ) = 0.402659566

Thus, the probability of at least 4 successes is

P(at least 4 ) = 0.597340434 [answer]

***************

c)

Using a cumulative binomial distribution table or technology, matching

n = number of trials = 12

p = the probability of a success = 0.33

x = the maximum number of successes = 2

Then the cumulative probability is

P(at most 2 ) = 0.187560896 [answer]

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Answer: 1/2 cup oats 2/3 cup flower

Answer:

tennis team

Step-by-step explanation:

because they ran 8 miles is 7 days to get equal amount of days you multiply by 2 on both 8 miles and 7 days so they ran 16 miles in 14 days thats y tennisbteam ran more miles

Answers in bold

7.  Not congruent

8.  Congruent by the HL theorem

HL = hypotenuse leg

==========================================

Explanation for problem 7

The tickmarks on segments MA and AT tell us that these segments are congruent. Another pair of congruent segments would be AE = AE by the reflexive property.

Then angle EMA = angle ETA because of the similar arc markings for those angles.

We have these congruence facts:

But recall that the "side side angle" rule isn't a valid congruence theorem. Search out "SSA ambiguous case" for more information.

We cannot use SAS because the angle is not between the sides mentioned.

Therefore, we don't have enough information to determine if the triangles are congruent or not. We can't say they are congruent, so the only thing we can do is say "not congruent" until more info comes along.

------------

Explanation for problem 8

We use the HL (hypotenuse leg) theorem. It works for right triangles only.

The congruent legs are segment ON = segment ON because of the reflexive property.

The congruent hypotenuses are MN = RO because of the tickmarks.

In short, the triangles are congruent by the HL theorem.

The amount of sand in each vase is 2.8 pounds by dividing the total amount by the number of vases.

Given that,

Raj combined white sand with black sand to make 14 pound mixture of sand.

Total amount of sand = 14 pounds

Number of vases to which sand is put = 5 vases

Amount of sand in each vase can be found by dividing the total amount by the number of vases.

Amount of sand in each vase = 14 / 5 = 2 4/5 pounds = 2.8 pounds

Hence the total amount of sand is 2.8 pounds.

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The surface area of the triangular prism is 70.2 yd²

A prism is a solid shape that is bound on all its sides by plane faces.

The surface area of a prism is expressed as ;

SA = 2B +ph

where h is the height,

B is the base area and

p is the perimeter of the base

Base area = 1/2 bh( since the base Is a triangle)

Base area = 1/2 × 3 ×3 = 4.5yd²

The other side of the triangle is calculated as;

x= √ 3²+3²

x = √9+9

x = √18

x = 4.2

Therefore perimeter = 4.2 + 3+3 = 10.2yds

SA = 2× 4.5 + 10.2 × 6

SA = 9 + 61.2

SA = 70.2 yd²

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The graph of the cubic equation is in the equation of the end.

To do it, we need to find some points that are solutions of the equation. Then we can graph these points on a coordinate axis and then connect these points with a curve proper of a cubic relation.

when x = 0

g(0) =  -2*0³-8*0² +18*0+72 = 72

So we have the point (0, 72)

when x = 1

g(1) =  -2*1³-8*1² +18*1+72

      = -2 - 8 + 18 + 72 = 80

So we have the point (1, 80)

when x = -1

g(-1) =  -2*-1³-8*-1² +18*-1+72

      =    2 - 8 - 18 + 72 = 48

(-1, 48)

And so on, when you have enough points, you can connect them. The graph that should you get is one like the graph in the image at the end.

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The expansion and simplification of the expression (x - 2)² is x² - 4x + 4.

An algebraic expression is a combination of variables with constants, numbers, and values using the mathematical operands addition, subtraction, multiplication, or division.

(x - 2)²

Expanding the square:

(x - 2)² = (x - 2)(x -2)

Distributing the square:

x(x - 2) - 2(x - 2)

x² - 2x - 2(x -2)

x² - 2x - 2x + 4

x² - 4x + 4

Thus, after expanding and simplifying the algebraic expression (x - 2)², the solution is x² - 4x + 4.

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Answer:

$1092

Step-by-step explanation:

The height of tree is given as follows:

40 ft.

The height of the tree is obtained applying the proportions in the context of this problem.

The proportions are applied as a rule of three can be formed between the heights and the shadows for both the tree and the person, as follows:

h/30 = 6/4.5

Applying cross multiplication, the value of h is obtained as follows:

4.5h = 30 x 6

4.5h = 180

h = 180/4.5

h = 40 ft.

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Based on Joey's debt-to-income ratio, the maximum monthly rent payment he can make and still maintain the DTI ratio would be; b. $898.

WE are given that Joey has a DTI ratio of 36% which means that his maximum debt payment should be 36% of his income.

This amount would be:

= DTI x monthly income

= 36% x 3,600

= $1,296

Maximum monthly rent can be found as:

= Maximum monthly debt payment - Car payment - Credit card payment

= 1,296 - 178 - 220

= $898

In conclusion, option B is correct.

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The velocity of the rock as a function of time is given by:

[tex]v(t) = 3 - 4.9exp(-2t/0.2) m/s[/tex]

where 4.9 is the value of mg in SI units.

The forces acting on the rock are the force due to gravity and the force due to air resistance. The force due to air resistance is given by 0.5v, where v is the velocity of the rock.

The force due to gravity is given by the mass of the rock (0.2 kg) times the acceleration due to gravity [tex](9.8 m/s^2)[/tex]. Using Newton's second law, we can set up the following differential equation:

[tex]m(dv/dt) = -mg - 0.5v[/tex]

where m is the mass of the rock, g is the acceleration due to gravity, and v is the velocity of the rock as a function of time t.

We can simplify this differential equation by dividing both sides by m:

[tex]dv/dt = (-g - 0.5v/m)v[/tex]

This is a separable differential equation, which we can solve using the separation of variables:

[tex](1/(-g - 0.5v/m)) dv = dt[/tex]

Integrating both sides gives:

[tex]-2ln(-g - 0.5v/m) = t + C[/tex]

where C is a constant of integration.

Solving for v gives:

[tex]v(t) = -0.5mg + C'exp(-2t/m)[/tex]

where C' = exp(C).

We can find the value of C' using the initial condition that the initial velocity of the rock is 3 m/s:

[tex]v(0) = -0.5mg + C' = 3[/tex]

[tex]C' = 0.5mg + 3[/tex]

Substituting this into the equation for v(t) gives:

[tex]v(t) = -0.5mg + (0.5mg + 3)exp(-2t/m)[/tex]

Therefore, the velocity of the rock as a function of time is given by:

[tex]v(t) = 3 - 4.9exp(-2t/0.2) m/s[/tex]

where 4.9 is the value of mg in SI units.

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The multiplying integers -2x24= -48

Here is some rules of multiplication of integers:

1. Positive integer × negative integer = negative.

2. Positive integers × Positive integers =  positive.

3.  Negative integers × Negative integers = positive.

Here, To find the  the multiplying integers

-2x24 = -48

When you multiply integers :

Negative x Positive = Negative

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The sides show that we have an equilateral triangle

An equilateral triangle is a triangle in which all three sides are of equal length. Since all three sides are equal, all three angles are also equal and measure 60 degrees each.

We know that;

12x - 22 = 10x - 6

Collect like terms;

12x - 10x = -6 + 22

2x = 16

x = 8

Thus the sides of the triangle are;

12(8) - 22 = 74

10(8) - 6 = 74

7(8) + 18 = 74

In the second triangle;

4x - 25 = x + 14

4x - x= 14 + 25

3x = 39

x = 13

Thus the sides are;

4(13) - 25 = 27

13 + 14 = 27

6(13) - 51 = 27

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Answer: x is greater than or equal to 5 (I can't put the symbol in)

Step-by-step explanation:

Add 5 to both sides then simplify, which will get you 2x is greater than or equal to 10, then divide by 2.

Answer:

Step-by-step explanation:

3x+10<3

3x<3-10

3x<-7

x<-7/3

2x-5≥5

2x≥5+5

2x≥10

x≥10/2

x≥5

so x<-7/3 or x≥5

The answers to the given questions about annual interest are given below:

a.

A = 4,000 (1 + 0.05/2)^(2 x 2)

  = $4,415.25

b.

A = 4,000 x e^(0.045 x 2)

  = $4,376.70

c. $100 more than option 2 = 4,376.70 + 100

                                             = $4,476.70

t (in years) = ln(4,476.70/4,000) /  ln(1 + 0.05/2)

                = 2.27986 years

Annual interest denotes the rate of interest levied or gained on a loan or investment for a duration of one year. This indicates the portion, expressed as a percentage, of the original amount that is utilized for interest payments or gained as a profit within a twelve-month period.

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Answer:

For this question, you can use the simultaneous equation to solve this problem.

Equation 1 reads: x + y = 34. (There are 34 notes in total.)

Equation 2: 5x + 10y = 235 (The notes are worth a total of $235.)

To find x in terms of y, we can apply equation 1:

x = 34 - y

When we use this expression to replace x in equation 2, we obtain:

5(34 - y) + 10y = 235

By condensing and figuring out y, we get at:

y = 15

There are 15 $10 bills in the wallet as a result.

Correct option is,

A: number of cars; money raised; 30 to 100 cars; $600 to $2000 is the right option.

Since, An independent variable is a variable that represents a quantity that is being controlled  in an experiment.

A dependent variable represents a quantity whose value depends on how the independent variable is controlled.

Now, In the given question;

The number of cars represents the independent variable while money raised represents independent variable .

Domain is the set of values the independent variable can take .

The number of cars 30 to 100 represents the Domain.

The range is corresponding y values .

The charge per car is $20.

Hence, Range = 30 x 20

                       =600

to Range = 100 x 20

              = 2000.

Thus, Correct option is,

A: number of cars; money raised; 30 to 100 cars; $600 to $2000 is the right option.

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Complete question is,

The French club is holding a car wash fundraiser. They are going to charge $20 per car, and expect between 30 and 100 cars. Identify the independent and dependent quantity in the situation, and find reasonable domain and range values.

A: number of cars; money raised; 30 to 100 cars; $600 to $2000

B: money raised; number of cars; 30 to 100 cars; $600 to $2000

C: number of cars; money raised; $600 to $2000; 30 to 100 cars

D: money raised; number of cars; $600 to $2000; 30 to 100 cars