Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that there is a positive integer that is not the sum of three squares.

Answer:

[tex]x=50000[/tex]

[tex]P(x)=0.7x-35000[/tex]

Step-by-step explanation:

Given cost function is

[tex]C(x)=0.85x+35000[/tex]

and revenue function is

[tex]R(x)=1.55x[/tex]

At break even point, revenue is equal to cost

R(x)= C(x)

[tex]1.55x=0.85x+35000[/tex]

Subtract 0.85 from both sides

[tex]0.7x=35000[/tex]

divide by 0.7 on both sides

[tex]x=50000[/tex]

Profit function

P(x)= R(x)- C(x)

[tex]P(x)=1.55x-(0.85x+35000)[/tex]

[tex]P(x)=1.55x-0.85x-35000[/tex]

[tex]P(x)=0.7x-35000[/tex]

The break-even point is found by setting the revenue function equal to the cost function, which results in the sale of 50,000 units. The profit function is calculated by subtracting the cost function from the revenue function, resulting in π(x) = 0.70x - 35,000.

To find the break-even point where the cost and revenue functions are equal, we substitute these functions and solve for Q:

R(x) = C(x)

1.55x = 0.85x + 35,000

This gives us 1.55x - 0.85x = 35,000

0.70x = 35,000

x = 35,000 / 0.70

x = 50,000 units (break-even point)

The profit function (π) is found by subtracting the cost function from the revenue function:

π(x) = R(x) - C(x)

π(x) = 1.55x - (0.85x + 35,000)

π(x) = 1.55x - 0.85x - 35,000

π(x) = 0.70x - 35,000

Hence, the profit function is π(x) = 0.70x - 35,000.

Answer:

Tara need to run 2 laps today.

Step-by-step explanation:

Number of laps in 1 mile = 3 laps

number of miles she wants to run today = [tex]\frac{2}{3}[/tex]

We need to find the number of lap she need to run today;

Solution:

Now we know that;

in 1 mile = 3 laps

In [tex]\frac{2}{3}[/tex] miles = number of laps in [tex]\frac{2}{3}[/tex] miles

By using Unitary method we get;

number of laps in [tex]\frac{2}{3}[/tex] miles = [tex]3\times\frac{2}{3} = 2\ laps[/tex]

Hence Tara need to run 2 laps today.

Answer:

The predicted height of the woman is 178.73 cm.

Step-by-step explanation:

Consider the provided function.

[tex]H(x) =2.75x+71.48[/tex]

Where x represents the height of humerus and H(x) represents the height of woman.

Substitute x=39 in above function.

[tex]H(x) =2.75(39)+71.48[/tex]

[tex]H(x) =107.25+71.48[/tex]

[tex]H(x) =178.73[/tex]

Hence, the predicted height of the woman is 178.73 cm.

The height of the woman is 178.73 cm.

Given to us,

function describing the height of a woman, H(x) =2.75x+71.48,

where, x is humerus (the bone from the elbow to the shoulder).

Height of a woman whose humerus is 39 cm long,

We can find the height of the woman by substituting the value of x in H(x).

H(x) =2.75x+71.48,

substituting x = 39 cm,

H(39) =2.75(39)+71.48,

         = 178.73 cm

Therefore,  the height of the woman is 178.73 cm.

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Final Answer:

The correct simplified ratio of US to British stamps is 25 : 2, which corresponds to answer choice E.

Explanation:

To determine the ratio of US to British stamps, we must use the given ratios of US to Indian and Indian to British stamps.

The given ratio of US to Indian stamps is 5 to 2. This means for every 5 US stamps there are 2 Indian stamps. We can represent this as:

US : Indian = 5 : 2

The given ratio of Indian to British stamps is 5 to 1. This implies that for every 5 Indian stamps, there is 1 British stamp. We can write this as:

Indian : British = 5 : 1

To find the ratio of US to British stamps, we need to combine these two ratios. To do this, we should express each ratio such that the Indian stamp part of each ratio is the same. Since the ratios already have the same number of Indian stamps (5 of them), we can directly multiply the US part by the British part across these ratios.

Thus, we multiply the number of US stamps (5 from the first ratio) by the number of British stamps (1 from the second ratio):

US to British = 5 (US) * 1 (British)
US to British = 5

Since we did not have to multiply the number of British stamps by anything (they were only multiplied by 1), the British part of the ratio remains unchanged.

Next, we multiply the Indian part of the first ratio by the British part of the second ratio:

Indian to British = 2 (Indian from the first ratio) * 5 (British from the second ratio)
Indian to British = 10

Now we can combine these two results to express the US to British ratio:

US to British = 5 (US) : 10 (British)

To simplify this ratio, we divide both sides by the common factor between them. In this case, that common factor is 5. So when we divide both numbers by 5, we have:

US to British = (5/5) : (10/5)
US to British = 1 : 2

However, this is not the option given in the question. Let's revisit the calculation and see if there was an error:

The correct approach to combining the ratios is to multiply the two ratios directly:

US : Indian = 5 : 2
Indian : British = 5 : 1

Multiplying across gives us:

US : British = (5 * 5) : (2 * 1)
US : British = 25 : 2
The correct simplified ratio of US to British stamps is 25 : 2, which corresponds to answer choice E.

The additional root is 6-i.

Here's why:

Conjugate Pairs for Rational Coefficients: When a polynomial with rational coefficients has a complex root of the form a + bi (where a and b are real numbers and i is the imaginary unit), its conjugate, a - bi, must also be a root. This ensures that the polynomial remains with rational coefficients.

Applying the Conjugate Pair Rule: In this case, we're given that 6 + i is a root. Therefore, its conjugate, 6 - i, must also be a root of the polynomial P(x) = 0.

Other Roots: The problem states that √7 is also a root. However, since it's a real number, it doesn't introduce any complex conjugate pairs.

Quartic Polynomial: A quartic polynomial has four roots in total. We've identified three of them: √7, 6 + i, and 6 - i. The fourth root could be either a real number or another complex conjugate pair, but the information provided is insufficient to determine its exact value.

Answer:

The answer to your question is

a) t₁ = 6 h

b) t₂ = 3h

Step-by-step explanation:

Data

v1 = 12 mph

v2 = 24 mph

total time = 9 h

Process

1.- Write equations to solve the problem

d₁ = v₁t₁    ------------------ Equation l

d₂ = v₂t₂

d₁ = d₂     because the distance is the same in both directions

t₁ = t₁

t₂ = 9 - t₁

d₂ = v₂(9 - t₁)  -------------- Equation 2

- Equal both equations

    v₁t₁ = v₂(9 - t₁)

- Substitute v₁ and v₂

   12t₁ = 24(9 - t₁)

- Solve for t₁

    12t₁ = 216 - 24t₁

    12t₁ + 24t₁ = 216

              36t₁ = 216

                  t₁ = 216 / 36

                  t₁ = 6 h

- Calculate t₂

t₂ = 9 - 6

t₂ = 3 h

Answer:

The equation use to find the value of R is [tex]55+R=110[/tex].

Step-by-step explanation:

Given:

Total Number of flowers = 110

Number of orange flowers = 55

Let the number of red flowers be represented by 'R'

We need to find the equation used to find the value of R.

Solution:

Now we know that;

Total Number of flowers is equal to sum of Number of orange flowers and Number of red flowers.

representing in equation form we get;

[tex]55+R=110[/tex]

Hence The equation use to find the value of R is [tex]55+R=110[/tex].

On Solving the above equation we get;

We will subtract both side by 55 using Subtraction property of equality.

[tex]55+R-55=110-55\\\\R=55[/tex]

Hence There are 55 red flowers in the garden.

When a fair dice is rolled twice, it has 36 possible outcomes. The sum of the numbers that amount to 7 can be got in 6 ways. Therefore, the probability of rolling a sum of 7 is 1/6.

When solving a probability question, the first step is to determine all the possible outcomes. Since a dice has 6 sides, when you roll it twice, you have 6x6 = 36 possible outcomes.

We are looking for the outcomes where the sum of the numbers is exactly 7. There are six (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

The probability is then calculated by taking the number of desired outcomes and dividing it by the total number of outcomes. So, the probability of rolling a sum of 7 with two dice is 6 / 36 = 1 / 6.

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The division equation is: [tex]\frac{x}{19} = 12[/tex]

228 goldfish can be kept

Solution:

Given that,

A pet store has 19 goldfish tanks

The store can place 12 fish in each tank

Let "x" be the number of gold fish that can be kept in tank

From given information,

Number of goldfish tanks = 19

Number of fish kept in 1 tank = 12 fish

We know that,

number of gold fish that can be kept in tank = Number of goldfish tanks x Number of fish kept in 1 tank

[tex]x = 19 \times 12[/tex]

[tex]\frac{x}{19} = 12[/tex]

Thus the division equation is found

On solving we get,

x = 19 x 12 = 228

Thus 228 goldfish can be kept

Algebraically, x ≤ 5π - 32 solves the inequality. Graphically, the solution lies in the shaded region below the line y = 8 - x and above y = 5(8 - π) on the coordinate plane.

Algebraic Approach:

To solve the inequality 8 - x ≥ 5(8 - π), begin by distributing 5 on the right side: 8 - x ≥ 40 - 5π. Next, isolate x by subtracting 8 from both sides: -x ≥ -5π + 32. Multiply both sides by -1, and reverse the inequality sign: x ≤ 5π - 32. This gives the solution for the inequality.

Graphical Approach:

Represent the functions y = 8 - x and y = 5(8 - π) on a graph. The point of intersection is the solution to the inequality. The line y = 8 - x is a downward-sloping line passing through the point (0, 8). The line y = 5(8 - π) is a horizontal line parallel to the x-axis at a height of 5(8 - π). The shaded region below the line y = 8 - x and above y = 5(8 - π) represents the solution to the inequality.

The question probable may be:

Solve the following inequality using both the graphical and algebraic approach: 8-x≥ 5(8-π)

Answer:

4

Step-by-step exp

You figure out what is 20 percent of 20 and that is 4.

Answer:

Step-by-step explanation:

The total number of magnets that Lisa and Bill made for the craft fair is 60.

They sold about 55% of the magnets. The number of magnets that they sold would be about

55/100 × 60 = 0.55 × 60 = 33

If Lisa says that they sold about 30 magnets, she is correct because if we round off 33 to the nearest ten, it would be 30 magnets.

If Bill says that they sold about 36 magnets, he is wrong because if we round off 36 to the nearest ten, it would be 40 magnets.

Step-by-step explanation:

In ten minutes she walked 1-0.5=0.5 miles.

60 minutes/10 minutes=6. So 0.5 miles ×6=3 miles per hour

Answer:

The larger sculpture is 10.0 feet tall

Step-by-step explanation:

The height of one sculpture (larger sculpture) is four times the height of the other sculpture (smaller sculpture)

Let the height of the larger sculpture be x and the height of the smaller sculpture be y

Therefore, x = 4y (y = 2.5 feet)

x = 4×2.5 feet = 10.0 feet

The larger sculpture is 10 feet tall.

To find the height of the larger sculpture, we can use the information given. We know that the smaller sculpture is 2.5 feet tall, and the larger sculpture is four times as tall. So, we can multiply the height of the smaller sculpture by 4.

Larger sculpture height = 2.5 feet × 4 = 10 feet

Therefore, the height of the larger sculpture is 10 feet.

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Answer:

Explanation:

The table that shows the pattern for this question is:

Time (year) Population

  0                   40

  1                     62

  2                    96

  3                   149

  4                   231

A growing exponentially pattern may be modeled by a function of the form P(x) = P₀(r)ˣ.

Where P₀ represents the initial population (year = 0), r represents the multiplicative growing rate, and P(x0 represents the population at the year x.

Thus you must find both P₀ and r.

1) P₀

  Using the first term of the sequence (0, 40) you get:

   P(0) = 40 = P₀ (r)⁰ = P₀ (1) = P₀

Then, P₀ = 40

 2) r

Take two consecutive terms of the sequence:

You can verify that, for any other two consecutive terms you get the same result: 96/62 ≈ 149/96 ≈ 231/149 ≈ 1.55

3) Model

Thus, your model is P(x) = 40(1.55)ˣ

4) Population of moose after 12 years

Answer:

bryan walked dogs for 5.8 hours

Step-by-step explanation:

if he had $52 and the walk per hour is $9

you have to divide 52/9

52/9 is 5.77777777777...

after you roud it to the nearest tenth that is 5.8

Bryan earned $9 per hour and had a $20 expense for flyers. After subtracting his expenses from his net income of $52, we can set up an equation to find the number of hours he worked, which is 8 hours.

Let h equal the number of hours Bryan walked dogs.

Calculate his earnings by multiplying h by the rate per hour, which is $9.

Subtract his expenses, which are $20 for the flyers, from his earnings to find his net income.

Set up the equation: 9h - 20 = $52.

Add $20 to both sides of the equation to isolate the term with h on one side: 9h = $52 + $20.

Simplify the right side of the equation: 9h = $72.

Divide both sides by 9 to solve for h: h = $72 ÷ 9.

Solve for h: h = 8.

Therefore, Bryan walked dogs for 8 hours to earn a net income of $52 after his $20 expense on flyers.

Answer:

[tex]\frac{7}{3} + \frac{e^2 - e}{2}[/tex]

Step-by-step explanation:

By definition:

Work done along the path is the line integral along that path denoted as:

Work Done = [tex]\int\limits^C {F} \, dr[/tex]

Note:  dr = dx i + dy j

Given that: [tex]F (x,y) = x^2 i + ye^x j[/tex]

F (x, y) dot product with dr = [tex] x^2 dx + ye^x dy[/tex]

Work done = [tex]\int\limits^C {(x^2 dx + ye^x dy)}[/tex]  ... Eq 1

Given that C: [tex]y = \sqrt{x-1}[/tex]

[tex]dy = \frac{dx}{2\sqrt{x-1} }[/tex]

Replace the value of y and dy in Eq 1

[tex]Work done =  \int\limits^C ({x^2 + \frac{e^x}{2} }) \, dx[/tex]

Limits of x are 1 to 2 respectively

[tex]Work done =  \int\limits^2_1 ({x^2 + \frac{e^x}{2} }) \, dx[/tex]

= [tex](\frac{x^3}{3} + \frac{e^x}{2})\limits^2_1[/tex]

Evaluate limits to obtain

Work Done = [tex]\frac{7}{3} + \frac{e^2 - e}{2}[/tex]

Answer: 95% of all people work between 7 hours and 9 hours per day and It is true.

Step-by-step explanation:

Since we have given that

Mean = 8 hours

Standard deviation = 0.5 hours

According to Empirical Rule,

at 95% confidence, it lies within 2 standard deviations  from the mean

so, lower value is given by

[tex]mean-2\times sd\\\\=8-2\times 0.5\\\\\=8-1\\\\=7\ hours[/tex]

upper value is given by

[tex]mean+2\times s.d\\\\=8+2\times 0.5\\\\=8+1\\\\=9\ hours[/tex]

Hence, 95% of all people work between 7 hours and 9 hours per day.

Therefore , it is true.

Final answer:

95% of people work between 7 and 9 hours per day according to the empirical rule of the normal distribution. The statement is true.

Explanation:

The distribution of the number of hours people spend at work per day is unimodal and symmetric with a mean of 8 hours and a standard deviation of 0.5 hours. Since the distribution is symmetric, and we are looking for the range that includes 95% of the distribution for a standard normal distribution, we can use the empirical rule. The empirical rule states that approximately 95% of the data in a normal distribution falls within two standard deviations of the mean.

To find the range, we calculate as follows:

Therefore, 95% of all people work between 7 and 9 hours per day. The statement is true.

Answer: (a) spuriousness relationship

Step-by-step explanation:

Spurious occurs between two variables that are actually caused by a third variable. Examples is like a number of teachers in region and number of people learn from college.

The total caloric content in the serving of fish can be calculated by adding the calories from protein (200 kcal) and the calories from fat (36 kcal). Therefore, the serving of fish contains a total of 236 kcal.

To calculate the total calories in a serving of fish, we need to add the caloric content of both the protein and the fat. The protein content of the fish is 50 g, and we know that protein has a caloric value of 4 kcal/g. Thus, the total caloric content from protein is 50 g x 4 kcal/g = 200 kcal. The fat content of this serving of fish is 4 g, and fat has a caloric value of 9 kcal/g. This makes the total caloric content from fat 4 g x 9 kcal/g = 36 kcal.

To find the total caloric content of the serving, we need to add together the calories from protein and fat. So, 200 kcal + 36 kcal = 236 kcal. Therefore, the serving of fish contains a total of 236 kcal.

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Answer:

6 minutes

Step-by-step explanation:

Let the volume of the sink be Xm^3

The rate of filling the sink by the first faucet is R1 while the rate of draining the faucet is -R2(negative since it’s draining)

R1 = x/2

R2 = -x/3 ( negative as it is draining)

The time it would take both of them working at the same rate to fill the sink is as follows:

x/(R1 - R2) = T

x/( x/2 - x/3) = T

x = T(x/2 - x/3)

x = T( (3x - 2x)/6)

x = T(x/6)

x = Tx/6

6x = Tx

T = 6 minutes

Answer:

The cities are 495 miles apart.

Step-by-step explanation:

Let x represent the distance between the two cities.

Let t represent the time taken by the plane to fly between the two cities.

The plane flew between two cities at 330 mph.

Distance = speed × time

Distance covered by the plane would be

330 × t = 330t

A car went the same distance at 55 mph. If the car took 7.5 hours longer, means that the time spent by the car would be (t + 7.5) hours and distance travelled would be

55(t + 7.5) = 55t + 412.5

Since the distance is the same, then

330t = 55t + 412.5

330t - 55t = 412.5

275t = 412.5

t = 412.5/275

t = 1.5

the distance between the two cities would be

1.5 × 330 = 495 miles

Answer:

Systematic sampling.

Step-by-step explanation:

The systematic sampling is the type of random sampling when the first unit is selected at random from k units and then every kth unit is selected. The k is known as sampling interval which is equal to the population size divided by sample size i.e. N/n.

In the given scenario a quality control manager start with 6th and then every 14th soup canssoup is selected. The sampling units can be selected as  6, 20, 34, 48, 62, 76... and so on. Here the value of k is 14. Thus, the given sampling is the systematic sampling.

Answer:

[tex]\sin \theta = \frac{y}r} = \frac{-1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = -\frac{\sqrt{5}}{5}\\\\\cos \theta = \frac{x}{r} = \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = -\frac{2\sqrt{5}}{5} \\\\\tan \theta = \frac{y}{x} = \frac{-1}{2} = -\frac{1}{2} \\\\\cot \theta = \frac{x}{y} = \frac{2}{-1} = -2\\\\\sec \theta = \frac{r}{x} = \frac{\sqrt{5}}{2} \\\\\csc \theta = \frac{r}{y} = \frac{\sqrt{5}}{-1} = -\sqrt{5}[/tex]

Step-by-step explanation:

First, we need to draw the terminal position of the given angle. To do so, we need to find a point that lies on the straight line [tex] x + 2y= 0, x\geq 0 [/tex]

If we choose [tex] x = 2 [/tex] (we can do so because of the condition [tex] x \geq 0 [/tex], which means that any positive value is suitable for [tex] x [/tex]), then we have

[tex] 2 +2y = 0\implies 2 = -2y \implies y = -1 [/tex]

Therefore, the terminal side of the angle [tex] \theta [/tex]  is passing through the origin and the point  [tex] (2,-1) [/tex] and now we can draw it.

The angle  [tex] \theta [/tex]  is presented below.

The distance of the point  [tex] (2,-1) [/tex] from the origin equals

[tex]r = \sqrt{2^2 + (-1)^2} = \sqrt{5}[/tex]

Now, we can determine the values of the six trigonometric function, by using their definitions.

[tex]\sin \theta = \frac{y}r} = \frac{-1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = -\frac{\sqrt{5}}{5}\\\\\cos \theta = \frac{x}{r} = \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = -\frac{2\sqrt{5}}{5} \\\\\tan \theta = \frac{y}{x} = \frac{-1}{2} = -\frac{1}{2} \\\\\cot \theta = \frac{x}{y} = \frac{2}{-1} = -2\\\\\sec \theta = \frac{r}{x} = \frac{\sqrt{5}}{2} \\\\\csc \theta = \frac{r}{y} = \frac{\sqrt{5}}{-1} = -\sqrt{5}[/tex]

Answer: It is called a sample statistic.

Step-by-step explanation:

Since we have given that

A researcher records the time it takes to complete a memory task in a sample of 25 participants. He finds that the average participant completed the test in 43 s.

The average time to complete this task is called sample statistic.

As we know that

Sample statistic is the quantity which we get from the sample taken from a any specified population for using this quantity to calculate the same.

In our case, average participant completed the test = 43 sec.

It is sample statistic as we will use to find the average time for the population i.e. 25 participant to get the same calculation i.e. average time.

We will get average time is also equal to 43 sec.

Hence, it is called a sample statistic.

We need more data about the box dimensions to calculate the savings. The formula involves multiplying the cost per square foot with the difference in surface areas and number of boxes. The idea of economies of scale isn't directly applicable in this context.

In order to answer this question, we would require additional information about the dimensions of the boxes and their surface areas. The cost difference between making boxes with different surface areas can be found by multiplying the difference in surface areas by the cost per square foot and then by the number of boxes.

The formula used would be: $1.10 (Cost per Square Foot) * Difference in Surface Areas * 900 (Number of Boxes).

In case of economies of scale, like the information provided about alarm clocks, the cost per box would decrease as the number of boxes produced increases. But this concept isn't directly applicable here as we're dealing with the material cost of the boxes, not the production cost.

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Answer:

  500 points

Step-by-step explanation:

Rocco scored 1/5 as many points as Stephen, so scored ...

  (1/5) × (2500 points) = 500 points

Rocco scored 500 points.

Purchasing the regular size is more economical because the regular size costs $​0.0369 less per gram.

Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.

It is given that:

The economy size of a detergent at 7 kilograms for ​$7.15 or purchase the regular size at 920 grams for 60 cents.

As we know.

1 kg = 1000 grams

7 kg = 7000 grams

$7.15 = 715 cents

7000 grams cost 715 cents

1 gram cost:

Per gram cost = 715/7000 = cent 0.1021 per gram

920 grams cost 60 cents

Per gram cost:

= 60/920

= cent 0.0652 per gram

Difference in cost = 0.0369

Purchasing the regular size is more economical because the regular size costs $​0.0369 less per gram.

Thus, purchasing the regular size is more economical because the regular size costs $​0.0369 less per gram.

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Answer:

Table 3

Step-by-step explanation:

The third one.

We have the function

                               [tex]h(x) = \sqrt[3]{-x+2}[/tex]

Now we will insert values of x in that definition o h(x) and see if the values we obtain match the corresponding y values in the table:

                  [tex]h(-6) = \sqrt[3]{-(-6)+2}= \sqrt[3]{6+2}= \sqrt[3]{8} = 2\\h(1) = \sqrt[3]{-1+2}= \sqrt[3]{1}= 1\\h(2) = \sqrt[3]{-2+2}= \sqrt[3]{0}= 0\\h(3) = \sqrt[3]{-3+2}= \sqrt[3]{1}= 1\\h(10) = \sqrt[3]{-10+2}= \sqrt[3]{-8}= -2[/tex]

We can see that the values match the table 3, so the table 3 represents points on the graph of h(x)

Answer:

C

Step-by-step explanation:

Final answer:

The student's question is about finding the probability that at least two of the 11 balls drawn with replacement from a box of 80 uniquely numbered balls are the same. We use the complement rule to find the probability of all balls being unique and subtract this from 1 to find the desired probability.

Explanation:

The student is asking about the probability of drawing at least two balls with the same number when 11 balls are drawn with replacement from a box containing 80 uniquely numbered balls. To solve this, we can use the complement rule, which states that the probability of an event occurring is equal to one minus the probability of the event not occurring.

The probability of drawing 11 unique balls in a row with replacement from 80 can be calculated by multiplying the probabilities of drawing a unique ball at each draw after the first. For the first ball, the probability is 80/80 (since any ball can be drawn), for the second ball it's 79/80 as one unique ball is already drawn, for the third it's 78/80, and so on until the eleventh ball.

The probability of drawing 11 unique balls is therefore (80/80) * (79/80) * (78/80) *...* (70/80). The probability of at least two balls having the same number is 1 minus this product, which represents the probability of all balls being unique.