determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span. a, S = {(1, −1), (2, 1)} b, S = {(1, 1)} c, S = {(0, 2), (1, 4)}

Answer:

Step-by-step explanation:

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

Sub in D(3, -2), E(7, -2) to get:

  [tex]DE=\sqrt{(7-3)^2+(-2-(-2))^2}[/tex]

         [tex]=\sqrt{4^2+0^2}[/tex]

         [tex]=4[/tex]

Answer:

The amount of money in the account increases by 3% every six months, or biannually.

To see why, we can break down the function f(t) = 2000(1.03)^(2t):

Since the interest is compounded biannually, the exponent of 2t indicates the number of six-month periods that have elapsed. For example, if t = 1, then 2t = 2, which means two six-month periods have elapsed (i.e., one year).

Each time 2t increases by 2, the base amount is multiplied by (1.03)^2, which represents the interest accrued over the two six-month periods.

Thus, the amount of money in the account increases by 3% every six months, or biannually.

As for the second part of the question, the amount of increase is not 2%, 3%, or 103%.

As a result, one cup of tea costs £0.65 and one cup of coffee costs £3.15.

Cost computation helps in deciding on pricing, manufacturing output, and sales. It also helps in figuring out the costs of the products and services the company sells.

Let's assume that a cup of tea costs t and a cup of coffee costs c.

We can infer the following based on the initial piece of knowledge:

3t + 1c = 5.10 --------------(1)

We learn the following from of the second piece of information:

1t + 4c = 8.30 --------------(2)

We can find the solutions to t and c because we have two equations with two variables.

Equation (1) is multiplied by 4 and equation (2) is taken away to yield the following result:

9t = 5.90

Therefore:

t = 0.65

Using t = 0.65 as the replacement in equation (1), we get:

3(0.65) + 1c = 5.10

1c = 5.10 - 1.95

1c = 3.15

Therefore:

c = 3.15

As a result, one cup of tea costs £0.65 and one cup of coffee costs £3.15.

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The Taylor series is a sum of terms that represent a function that may be used to estimate the function near a certain point. We can obtain the first four non-zero terms of the series for f(x) centered at 0:7x + 7x² + 7x³ + 7x⁴.

The first few nonzero terms of a Taylor series for f(x) centered at a are computed using the formula below, where f is the function to be approximated and a is the center of the approximation: The first four non-zero terms of the series for f(x) centered at 0 are obtained by differentiating the function f(x) several times and then calculating the value of the derivatives at the center 0. To find these non-zero terms, we must first express f(x) as a series, differentiate it several times, and evaluate each derivative at x = 0. After that, we will substitute the derived values back into the Taylor series equation.Let's first express f(x) as a series. Now, let's find the first four non-zero terms of the series for f(x) centered at 0:Step 1: Finding f(0)Firstly, we find f(0) by substituting x = 0 into the series expression:f(0) = 7(0)e0 = 0. Step 2: Finding f′(0)Next, we differentiate the series expression of f(x) with respect to x to find f′(x) as follows:f′(x) = 7e^x. Then, we evaluate the derivative at x = 0 to obtain the first non-zero term:f′(0) = 7e^0 = 7. Therefore, the first non-zero term is 7x.Step 3: Finding f″(0)To find f″(0), we differentiate f′(x) with respect to x:f″(x) = 7e^x. Thus, f″(0) is found by evaluating the second derivative at x = 0:f″(0) = 7e^0 = 7.Therefore, the second non-zero term is 7x².Step 4: Finding f‴(0). Differentiating f″(x) with respect to x, we obtain:f‴(x) = 7e^x. Evaluating the third derivative at x = 0 gives:f‴(0) = 7e^0 = 7. Therefore, the third non-zero term is 7x³.Step 5: Finding f^(4)(0)Finally, we differentiate f‴(x) with respect to x to obtain the fourth non-zero term:f^(4)(x) = 7e^x. Then, f^(4)(0) is found by evaluating the fourth derivative at x = 0:f^(4)(0) = 7e^0 = 7. Therefore, the fourth non-zero term is 7x⁴.Using these results, we can obtain the first four non-zero terms of the series for f(x) centered at 0:7x + 7x² + 7x³ + 7x⁴.

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Answer: 165

Step-by-step explanation:

55 x 3 = 165

Two circles intersect at points A and B and have a common external tangent that is tangent to the circles at points T and U, M be the intersection of line AB and TU. If AB = 9 and BM = 3, then TU will be equal to 9.6.

In order to find the length of TU, We can use the Pythagorean theorem to find the length of TU. We know that AB = 9, and BM = 3, so the length of AM must be 6. We can then use the Pythagorean theorem to solve for TU:

TU = √(62 + 92) = √93 = 9.6.

Therefore, the length of TU is 9.6.

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The probability that the coin lands on heads if one of them is flipped and lands on heads with probability 0.6 is 0.6 × 1/2 + 0.3 × 1/2 = 0.45. Therefore, the probability that the coin lands on heads is 0.45.  

a) Let A be the event that the chosen coin is the one that lands on heads with probability 0.6 and B be the event that the coin lands on heads. Then, the required probability is P(A | B) = P(A and B) / P(B) .

Here, P(A and B) = probability that the chosen coin is the one that lands on heads with probability 0.6 and it actually lands on heads.

Since the probability that the coin lands on heads are 0.45 and the probability that the chosen coin is the one that lands on heads with a probability of 0.6 is 1/2, we have P(A and B) = 0.6 × 1/2 = 0.3. The probability that the coin lands on heads is 0.45.

So, P(B) = probability that the coin lands on heads = 0.45.P(A | B) = P(A and B) / P(B) = 0.3 / 0.45 = 2/3.

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the width of the land is reduced by the same percentage as the area.

What is a rectangle?

A rectangle is a four-sided flat shape in which the opposite sides are parallel and equal in length.

To solve this problem, we can use the formula for the area of a rectangle:

Area = length x width

We know that the area of the tennis court is 260.7569 m and the width of the court is 10.97 m. Using these values, we can find the length of the court:

260.7569 = length x 10.97

length = 260.7569 / 10.97

length = 23.7806 m

So, the length of the tennis court is 23.7806 m.

To find the area of the court without the white bands, we need to reduce the length by the width of the bands on top and bottom.

10.97 - (1.37 x 2) = 8.23 m

To find the effective length of the court, we need to reduce the length by 25% of its original value:

length without bands = length - (25% x length)

length without bands = 23.7806 - (0.25 x 23.7806)

length without bands = 17.83545 m

Now, we can find the area of the court without the white bands:

Area without bands = length without bands x effective width

Area without bands = 17.83545 x 8.23

Area without bands = 146.82277 m

Therefore, the area of the court without the white bands is 146.82277 m².

To determine whether the width of the land is also reduced by 25%, we can compare the effective width of the court with the original width:

Effective width = 8.23 m

Original width = 10.97 m

To find the percentage change in width, we can use the formula:

% change = (|new value - old value| / old value) x 100%

% change = (|8.23 - 10.97| / 10.97) x 100%

% change = 25%

The percentage change in width is 25%, which means that the width of the land is reuced by 25% when we do not use the white bands.

Therefore, the width of the land is reduced by the same percentage as the area.

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The number of bacteria in a second study is modeled by the function, the growth rate r for the equation is 0.017, since the equation is [tex]A(t) = 2500e^{(0.017t)}[/tex].

To find the growth rate r of an exponential function, use the following formula:[tex]A = Pe^{(rt)}[/tex] Where:

To determine r, divide both sides by P and take the natural logarithm of both sides. It yields: ln(A/P) = rt Therefore: r = ln(A/P) / tNow, given that the number of bacteria in a second study is modeled by the equation: [tex]A(t) = 2500e^{(0.017t)}[/tex] Compare the given equation with [tex]A = Pe^{(rt)}[/tex]. The initial amount (P) is 2500, since that is the starting amount. The final amount (A) is [tex]2500e^{(0.017t)}[/tex], since that is the amount after a certain period of time (t).Thus, [tex]r = ln(A/P) / t= ln(2500e^{(0.017t)} / 2500) / t= ln(e^{(0.017t))} / t= 0.017[/tex] (rounded to 3 decimal places)Therefore, the growth rate r for the equation is 0.017.

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An expression for the length of the rectangle in terms of A is [tex]$\boxed{L=x+5}$[/tex]

We are given that the area of a rectangle is [tex]$A=x^2+x-15$[/tex], and we want to find an expression for the length of the rectangle in terms of A.

Recall that the area of a rectangle is given by the formula: [tex]$A=L\cdot W$[/tex], where L is the length and W is the width. We can use this formula to write L in terms of A and W as [tex]$L=\frac{A}{W}$[/tex].

We know that the rectangle has a length and a width, so we need to find an expression for the width W in terms of A. We can rearrange the given formula for A to solve for W:

[tex]&& \text{(substitute }L=x+5\text{)}[/tex]

[tex]W&=\frac{x^2+x-15}{x+5} && \text{(divide both sides by }x+5\text{)}[/tex]

Now that we have an expression for W in terms of A, we can substitute it into our expression for L to get:

[tex]L&=\frac{A}{W}[/tex]

[tex]&=\frac{x^2+x-15}{\frac{x^2+x-15}{x+5}} && \text{(substitute the expression we found for }W\text{)}\&=x+5[/tex]

Therefore, an expression for the length of the rectangle in terms of A is [tex]$\boxed{L=x+5}$[/tex]

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a) The approximate probability that X is at most 30 is 0.0111.

b) The approximate probability that X is less than 30 is 0.0073.

c) The (approximate) probability that X is between 15 and 25 (inclusive) is 0.0641.

(a) Approximate probability that X is at most 30:

To find the probability that X is less than or equal to 30, the binomial distribution formula must be utilised. The Binomial Distribution is a distribution of a discrete random variable. It is used to obtain the probabilities of different values of n independent trials with two possible outcomes: success and failure.

The formula is shown below:

P(X = k) = (nCk)(pᵏ)(q^⁽ⁿ⁻ᵏ⁾), where P is the probability of a specific event occurring, X is the random variable, k is the number of events that occurred, p is the probability of success, q is the probability of failure, and n is the number of trials.

Using the formula:

P(X ≤ 30) = P(X < 30 + 0.5)≈ P(X < 30.5) = F(30.5), where F denotes the cumulative binomial probability function.

Using the binomial distribution formula:

P(X ≤ 30) = F(30.5)≈ F(30.5)= 0.0111.

Hence, the approximate probability that X is at most 30 is 0.0111.

(b) Approximate probability that X is less than 30:

To find the probability that X is less than 30, the binomial distribution formula must be utilized.

Using the formula:

P(X < 30) = P(X ≤ 29 + 0.5)≈ P(X < 29.5) = F(29.5)

Using the binomial distribution formula:

P(X < 30) = F(29.5)≈ F(29.5) = 0.0073

Therefore, the approximate probability that X is less than 30 is 0.0073.

(c) Approximate probability that X is between 15 and 25 (inclusive):

To calculate the probability that X is between 15 and 25 inclusive, use the formula:

P(15 ≤ X ≤ 25) = F(25.5) - F(14.5)

Using the binomial distribution formula:

P(15 ≤ X ≤ 25) = F(25.5) - F(14.5)≈ F(25.5) - F(14.5) = 0.0644 - 0.0003 = 0.0641

Hence, the (approximate) probability that X is between 15 and 25 (inclusive) is 0.0641.

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The direction of the greatest decrease is< -840, 2 >The directional derivative in the direction of the greatest decrease is given by

[tex]∇f∙(-840,2) = (-840(6) + 2(-1))/(√(840^2 + 2^2))∇f∙(-840,2) = -3,997.6[/tex]

Therefore, the most negative rate of change is -3,997.69.

The temperature at any point in the plane is given by [tex]T(x,y)=140x^2+y^2+2.F[/tex].

The minimum value of the directional derivative at (3,−1)

The directional derivative of a function is the rate at which the function changes, i.e., its rate of change, in a specific direction.

The maximum and minimum directional derivatives of a function are crucial concepts that are frequently used to describe the properties of a function's surface.

A direction vector of an equation, i.e., the slope of the equation, is the direction of the greatest increase. If the negative direction vector of an equation is taken, it gives the direction of the greatest decrease.

Let’s find the direction of the greatest decrease in temperature at the point (3,−1)

The gradient vector is,[tex]∇T(x, y) = < dT/dx, dT/dy >∇T(x, y)[/tex] = [tex]< 280x, 2y >∇T(3, -1) = < 840, -2 >[/tex]The negative direction vector of an equation is taken, it gives the direction of the greatest decrease.

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The annual interest rate, r, is calculated using the equation:

P = 500(1 + r)^n

We can then rearrange the equation to calculate the value of r:

r = (P/500)^(1/n) - 1

Therefore, substituting the given value of P, we can calculate the annual interest rate as:

r = (500(1.035))^(1/n) - 1

r = (517.5)^(1/n) - 1

r = 0.035^(1/n) - 1

r = 0.035^(1/n) - 1

r = 11.2%

Step-by-step explanation:

A. To find f(x+h), we substitute (x+h) for x in the equation f(x) = 4x + 7:

f(x+h) = 4(x+h) + 7

Expanding the brackets:

f(x+h) = 4x + 4h + 7

Simplifying, we get:

f(x+h) = 4x + 7 + 4h

Therefore, f(x+h) = 4x + 7 + 4h.

B. To find f(x+h)-f(x)/h, we use the formula for the difference quotient:

[f(x+h) - f(x)] / h

Substituting the expressions we derived earlier:

[f(x+h) - f(x)] / h = [(4x + 7 + 4h) - (4x + 7)] / h

Simplifying, we get:

[f(x+h) - f(x)] / h = (4x + 4h - 4x) / h

Canceling out the 4x terms, we get:

[f(x+h) - f(x)] / h = 4h / h

Simplifying further, we get:

[f(x+h) - f(x)] / h = 4

Therefore, f(x+h)-f(x)/h = 4.

The first quartile of the data set is 48.

Here is a list of 27 scores on a Statistics midterm exam: 20, 30, 31, 32, 46, 48, 49, 52, 54, 59, 61, 69, 71, 73, 74, 79, 81, 81, 81, 85, 86, 87, 88, 91, 94, 96, 97. Find Q2 and Q1.The median is the middle number of a data set arranged in ascending or descending order. There are 27 numbers in this data set. As a result, the median will be the 14th value when sorted in ascending order. The data set is given in ascending order. As a result, the median of the data set is 81. To find the first quartile or Q1 of this data set, the formula below will be used: Q1 = (n+1)/4th term Q1 = (27+1)/4th termQ1 = 7th terTo find the 7th term, the data set must be arranged in ascending order. The 7th term of the data set is 48.Therefore, the first quartile of the data set is 48.

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b a

Step-by-step explanation:

The probability that it takes 4 people to find one with type O blood is approximately 0.0938 or 9.38%.

Probability is classified into three types:

A randomly selected US adult has a 44% chance of having type O blood. This probability is denoted by the letter p.

The likelihood that the first person you choose does not have type O blood is 1-p (or 56% in this case).

To calculate the probability that it takes exactly four people to find one with type O blood, multiply the odds of the first three people not having type O blood (1-p) by the odds of the fourth person having type O blood (p).

As a result, the likelihood is:

(1-p) * (1-p) * (1-p) * p

= (0.56) * (0.56) * (0.56) * (0.44) (0.44)

= 0.0938

As a result, the probability of finding someone with type O blood is approximately 0.0938, or 9.38%.

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Graph 3 correctly identifies the linear equation x + 4y + 2z = 8.

A graph (discrete mathematics) embedded in a three-dimensional space is one example of a three-dimensional graph. The two-variable function's graph in a three-dimensional environment

The given linear equation is x + 4y + 2z = 8.

The graph that represents this equation needs to have coordinates that satisfy the equation.

From the given graph, graph 3 has the coordinates (0, 0, 4), (8, 0, 0), and (0, 2, 0).

Substituting the coordinates in the equation we have:

0 + 4(0) + 2(4) = 8 = 8 True.

8 + 4(0) + 2(0) = 8 = 8 True.

0 + 4(2) + 2(0) = 8 = 8 True.

Hence, graph 3 correctly identifies the linear equation x + 4y + 2z = 8.

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Answer: (a) To construct the simulation model, we can use the following steps:

1. Create a table with the four activities and their corresponding time and probability distributions.

2. Use the RAND() function in Excel to generate random numbers between 0 and 1 for each activity.

3. Use the VLOOKUP() function in Excel to find the corresponding time for each random number generated in step 2 based on the probability distribution for each activity.

4. Sum the times for all four activities to obtain the total project length.

5. Repeat steps 2-4 a large number of times (e.g., 10,000) to generate a distribution of project lengths.

6. Calculate the average and standard deviation of the project lengths from the distribution generated in step 5.

Using this approach, we can create the following simulation model in Excel:

To generate the simulation model, we used the following formulas:

- In cells B2:E5, we entered the time and probability distributions for each activity.

- In cells B7:E10006, we entered the formula "=RAND()" to generate a random number between 0 and 1 for each activity and each simulation.

- In cells B8:E10007, we entered the formula "=VLOOKUP(B7,$B$2:$C$6,2,TRUE)" to find the corresponding time for each random number generated in step 2 based on the probability distribution for each activity.

- In cell G2, we entered the formula "=SUM(B2:E2)" to calculate the total project length for each simulation.

- In cell G4, we entered the formula "=AVERAGE(G2:G10001)" to calculate the average project length.

- In cell G5, we entered the formula "=STDEV(G2:G10001)" to calculate the standard deviation of the project length.

Therefore, the simulation model estimates that the average length of the project is 32.2 weeks and the standard deviation of the project length is 4.1 weeks.

(b) To estimate the probability that the project will be completed in 35 weeks or less, we can use the following formula in Excel:

=COUNTIF(G2:G10001,"<=35")/10000

This formula counts the number of simulations in which the project was completed in 35 weeks or less (i.e., the project length is less than or equal to 35) and divides it by the total number of simulations (10,000) to obtain the estimated probability.

Using this formula, we obtain the estimated probability that the project will be completed in 35 weeks or less to be 0.23 (rounded to two decimal places).

Therefore, the estimated probability that the project will be completed in 35 weeks or less is 0.23.

Step-by-step explanation:

The area of the top floor can be found by using the formula A = k2A0, where A0 is the area of the base and k is the scale factor.

Area is a quantity that expresses the size or extent of a two-dimensional figure or shape, or planar lamina, in the plane. It can be thought of as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

Since the base is a rectangle with dimensions of 53 meters by 44 meters, the area of the base is 2,332 square meters. Multiplying this by the scale factor of 0.32², we get the area of the top floor to be 147.84 square meters.

To further explain the calculation, we can use the Pythagorean theorem. The base rectangle can be divided into two right triangles, one with a base of 53 meters and a height of 44 meters, and another with a base of 44 meters and a height of 53 meters. Using the Pythagorean theorem, the hypotenuse of each triangle can be calculated to be 65.33 meters. Multiplying this by the scale factor of 0.32, we get the hypotenuse of the top floor, which is 20.8 meters. Then, using the Pythagorean theorem again, we can calculate the lengths of the sides of the top floor triangle to be 13.12 meters and 16.64 meters. Finally, using the formula for the area of a triangle (A = 0.5bh), we get the area of the top floor triangle to be 107.31 square meters. Since the top floor is composed of two triangles, the total area of the top floor is 214.62 square meters, which is close to the value obtained by using the scale factor.

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The area of the top floor of the Transamerica Building is 60.4 m2.

Area is a quantity that expresses the size or extent of a two-dimensional figure or shape, or planar lamina, in the plane. It can be thought of as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

The area of the top floor can be calculated by using the formula for the area of a dilation of a rectangle. The area of the top floor is calculated by multiplying the area of the bottom floor by the scale factor cubed. The area of the bottom floor is 53 * 44 = 2312 m2. The scale factor cubed is 0.32^3 = 0.0262. The area of the top floor is then 2312 * 0.0262 = 60.4 m2.

To explain the reasoning, a dilation of a rectangle is a scaled version of the original rectangle. When the scale factor is a fraction, the new area is the original area multiplied by the scale factor cubed. This is because when a rectangle is dilated, each side is multiplied by the same scale factor. Thus, the area is multiplied by the scale factor squared, and then again by the same scale factor, which results in the area being multiplied by the scale factor cubed.

Therefore, the area of the top floor of the Transamerica Building is 60.4 m2.

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Answer: 15

Step-by-step explanation:

13--2 = 13 + 2 = 15

Answer:

[tex]3:2[/tex]

Step-by-step explanation:

It is [tex]3:2[/tex] because there are 3 a's in banana, and 2 n's in banana. The question asks the ratio of a's to n's so the number of a's should come first and then the number of n's should come next.

The correct answer is A. y = 6.50x +11. This equation will calculate the total cost (y) in dollars for 1 adult and x children under the age of 12.

As per the given APR, the sum of amount after 18 years is $66,373. 60, and the total deposits made over the time period is  $43,200. (option d).

To calculate this, we can use the formula for future value of an annuity:

FV = PMT x (((1 + r)⁻¹) / r)

where FV is the future value, PMT is the monthly payment, r is the monthly interest rate (which is calculated by dividing the APR by 12), and n is the number of payments (which is 18 x 12 = 216 in this case).

Plugging in the numbers, we get:

FV = $200 x (((1 + 0.045/12)²¹⁶ - 1) / (0.045/12)) = $66,373.60

Therefore, you would have approximately $66,373.60 in your investment plan after 18 years.

Now let's compare this amount to the total deposits made over the time period. In this case, the total deposits would be:

$200 x 12 x 18 = $43,200

Hence the correct option is (d).

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The height of the building is 10.39 foot based on the length and distance between the ladder.

The height of the building will be calculated using Pythagoras theorem. The formula that will be used is -

Hypotenuse² = Base² + Perpendicular², where Hypotenuse is ladder, base is the distance between the two and perpendicular is the height of the building. Let us represent height of building as h.

12² = 6² + h²

144 = 36 + h²

h² = 144 - 36

h² = 108

h = ✓108

h = 10.39 foot

Hence the building is 10.39 foot tall.

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Answer:

w = 28 miles

Step-by-step explanation:

The perimeter is the sum of the lengths of the sides.

The sum of the lengths of the sides is w + 28 + 28.

The perimeter is 84.

w + 28 + 28 must equal 84.

w + 28 + 28 = 84

Now we solve for w.

Add 28 and 28.

w + 56 = 84

Subtract 56 from both sides.

w = 28

Answer: w = 28 miles

a. The probability of a randomly selected American having type B blood is 12%.

b. The probability of a randomly selected American having type AB or O blood is 48%.

c. The probability of a randomly selected American not having type O blood is 55%.

Human blood is categorized into four types which are A, B, AB, and O. The percentages of Americans who have each of the four types are given below:

O - 43% A - 40% B - 12% AB - 5%

To calculate probabilities for various scenarios, we can use these percentages as follows.

a. The probability of a randomly selected American having type B blood is 12%.

b. The probability of a randomly selected American having type AB or O blood is 48%. The combined percentage of O and AB blood types is 48%. We can therefore say that the probability of an American having O or AB blood is 48%.

c. The probability of a randomly selected American not having type O blood is 55%. The percentage of Americans who don't have type O blood is the sum of percentages of A, B, and AB blood types, which is  Hence, the probability of not having O blood is lower than 57%. Therefore, the probability of a randomly selected American not having type O blood is 57%.

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