Chloe has enough sand to fill a rectangular sandbox with an area of
36 square units. She wants the outer edges of the sandbox to use as little material as possible.

Answer:

  (c)  Reflection across x = -2

Step-by-step explanation:

You want to know the line of reflection that causes VWYZ to be mapped to V'W'Y'Z' as shown in the attached figure.

The line of reflection is the perpendicular bisector of the segment joining a point and its image.

The midpoint of VV' is ((-11, 2) +(7, 2))/2 = (-4, 4)/2 = (-2, 2). The line VV' has the same y-coordinate at both ends, so is a horizontal line. Its perpendicular bisector is the vertical line through (-2, 2), which is the line ...

  x = -2 . . . . . choice C

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Probability is a measure of the likelihood of an event occurring.

A. To find how many foreclosures to expect, we can use the expected value formula: E(X) = np, where X is the number of foreclosures, n is the number of loans, and p is the foreclosure rate. Substituting the given values, we have E(X) = (300)(1/136) ≈ 2.21. Therefore, we can expect about 2 foreclosures last month.

B. To find the probability of exactly two foreclosures, we can use the Poisson distribution with parameter λ = E(X) = 2.21. The probability mass function of the Poisson distribution is given by P(X = k) = (e^(-λ) * λ^k) / k!, where k is the number of foreclosures. Substituting k = 2 and λ = 2.21, we have P(X = 2) = (e^(-2.21) * 2.21^2) / 2! ≈ 0.244. Therefore, the probability of exactly two foreclosures is about 0.244.

C. To find the probability of at least one foreclosure, we can use the complement rule: P(X ≥ 1) = 1 - P(X = 0). Since the foreclosure rate is one out of every 136 loans, the probability of not having a foreclosure in a single loan is (135/136). Therefore, the probability of not having a foreclosure in 300 loans is (135/136)^300 ≈ 0.427. Substituting this value into the complement rule, we have P(X ≥ 1) = 1 - 0.427 ≈ 0.573. Therefore, the probability of at least one foreclosure is about 0.573.

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Therefore, the 90% confidence interval for μ1 - μ2 is -1 ± 1.74 years, which means that we are 90% confident that the true difference in mean ages between registered Republicans and Democrats is between -2.74 years and 0.74 years.

To find the confidence interval for the difference of means, we can use the following formula:

(X1 - X2) ± t*√(s1²/n1 + s2²/n2)

where X1 and X2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and t is the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom and the desired level of confidence.

We are given:

X1 = 39 years

s1 = 8 years

n1 = 128

X2 = 40 years

s2 = 10 years

n2 = 200

90% confidence level

First, we need to calculate the pooled standard deviation:

sp = √(((n1-1)*s1² + (n2-1)*s2²) / (n1+n2-2))

sp = √(((128-1)*8² + (200-1)*10²) / (128+200-2))

sp = 9.29

Next, we calculate the t-value:

t = invT(0.05, 327)

t = 1.66

where invT is the inverse t-distribution function with (n1+n2-2) degrees of freedom and a one-tailed area of 0.05 (since we want a 90% confidence interval, which corresponds to a one-tailed area of 0.05).

Finally, we can calculate the confidence interval:

(X1 - X2) ± T√(s1²/n1 + s2²/n2)

(39 - 40) ± 1.66√(8²/128 + 10²/200)

-1 ± 1.74

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Complete question:

A political analyst was curious if younger adults were becoming more conservative. He decided to see if the mean age of registered Republicans was lower than that of registered Democrats. He selected an SRS of 128 registered Republicans from a list of registered Republicans and determined the mean age to be = 39 years, with a standard deviation s1 = 8 years. He also selected an independent SRS of 200 registered Democrats from a list of registered Democrats and determined the mean age to be = 40 years, with a standard deviation s2 = 10 years. Let μ1 and μ2 represent the mean ages of the populations of all registered Republicans and Democrats, respectively. Suppose that the distributions of age in the populations of registered Republicans and of registered Democrats have the same standard deviation. Assume the pooled two-sample t procedures are safe to use. A 90% confidence interval for μ1 – μ2 is: –1 ± 1.66 years. 1 ± 2.76 years. –1 ± 1.74 years. –1 ± 1.98 years.

The values of angle N in the triangle is 71.3 degrees

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

l = 6.9 cm, m = 8.1 cm and n=8.8

Using the law of cosine, we have

n^2 = l^2 + m^2 - 2lmCos(N)

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

8.8^2 = 6.9^2 + 8.1^2 - 2 * 6.9 * 8.1Cos(N)

So, we have

- 2 * 6.9 * 8.1Cos(N) = -35.78

This gives

Cos(N) = 0.32

Evaluate and take the arc cos

N = 71.3 degrees

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Answer:

207.345

Step-by-step explanation:

Formula: A=2πrh+2πr2

Work : 2·π·3·8+2·π·32≈207.34512

Answer:  207.351 [tex]m^{2}[/tex]

Step-by-step explanation:

Surface Area is = Lateral Area + 2(Base Area)

LA = Ph       P, perimeter = C= 2[tex]\pi r[/tex]    r = 3m

                                           P= 2[tex]\pi 3[/tex]

                                            P= 18.849

                     h=8

LA = 18.849(8)

   =150.7964  This is the lateral area

B= [tex]\pi r^{2}[/tex]   Trying to find the Base area from original formula

=[tex]\pi 3^{2}[/tex]

=28.274

Put it all together

SA = LA+2B

=150.345+2(28.274)

=207.351 [tex]m^{2}[/tex]

The area of the composite figure is equal to 294.491 square meters.

In this problem we find the representation of a composite figure formed by a circular sector and an isosceles triangle. The area formulas for each case are shown below:

Circular sector

A = π · (α / 360) · r²

Triangle

A = 0.5 · (2 · r · sin 0.5α) · (r · cos 0.5α)

A = r² · sin 0.5α · cos 0.5α

Where:

A = π · (230 / 360) · (11.1 m)² + (11.1 m)² · sin 65° · cos 65°

A = 294.491 m²

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Hey ⊂hcpsibekweou⊃

Answer:

( 6 , 2 )

Step-by-step explanation:

Based on the image we can see that P' is in quadrant 4.

Coordinate plane:

As you may know reflection is known as a flip.

For example: (-5, 4) in quadrant 1  reflects to ( -5, -4 ) quadrant 3.

From the given we can see that P' is (6, -2 ),    Therefore the reflection of             ( 6, - 2) is ( 6 , 2 ).

xcookiex12

4/19/2023

In this scenario, we are interested in testing a hypothesis about the proportion of first-year college students who are undecided about their major. The claimed proportion is 0.47, but the researcher believes this value is too high. To test this hypothesis, the researcher surveys a random sample of 450 first-year college students and finds that 189 of them are undecided about their major.

To find the p-value in this problem, we need to conduct a hypothesis test. Let's define the terms first:

1. Proportion: The proportion of first-year college students who are undecided about their major, claimed to be 0.47.
2. Hypothesis: We are testing if the claimed proportion (0.47) is too high.
3. Sample: A random sample of 450 first-year college students, with 189 undecided about their major.

Now, let's follow the steps of a hypothesis test:

Step 1: Define the null hypothesis (H0) and alternative hypothesis (H1).
H0: p = 0.47 (The proportion is equal to 0.47)
H1: p < 0.47 (The proportion is less than 0.47)

Step 2: Calculate the test statistic.
Test statistic = (Sample proportion - Claimed proportion) / Standard error
Sample proportion = 189 / 450 = 0.42
Standard error = sqrt[(0.47 * (1 - 0.47)) / 450] ≈ 0.0229

Test statistic = (0.42 - 0.47) / 0.0229 ≈ -2.18

Step 3: Find the p-value using the test statistic from a Z-distribution table or calculator.
The test statistic is -2.18, which corresponds to a p-value of approximately 0.0146.

Step 4: Compare the p-value to the chosen significance level (alpha).
The p-value is between 0.01 and 0.05.

Therefore, the answer is: C. between 0.01 and 0.05.

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To organize the information about summer jobs available for college students in my area, I could use a spreadsheet as my data set.

(a) The cases would be each job opportunity available for college students in the area.

(b) The variables would include:
- Job Title
- Company Name
- Job Description
- Pay Rate
- Location
- Required Skills/Qualifications

The possible values for each variable would depend on the specific jobs available.

(c) The variables "Job Title," "Company Name," "Location," and "Required Skills/Qualifications" would be categorical variables, as they involve categories or labels. The variables "Job Description" and "Pay Rate" would be quantitative variables, as they involve numerical values.

(d) I would label this data set as "Summer Jobs for College Students in [Name of Area]." I chose this label to clearly indicate the focus of the data set.

(e) The key characteristics of this data set would include the job title, company name, job description, pay rate, location, and required skills/qualifications for each summer job available for college students in the area. By organizing this information, students can compare and contrast job opportunities to find the best fit for their skills and needs.

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The slope of the tangent line at (4, 54) is 45. So the equation of the tangent line to the curve y = x^3 - 3x + 2 at the point (4, 54) is y = 45x - 126.

To find the equation of the tangent line to the curve y = x^3 - 3x + 2 at the point (4, 54), we need to use calculus. First, we find the derivative of the function:

y' = 3x^2 - 3

Next, we plug in x = 4 to find the slope of the tangent line at that point:

y'(4) = 3(4)^2 - 3 = 45

So the slope of the tangent line at (4, 54) is 45. To find the equation of the line, we use the point-slope form of the equation:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the point on the line. Plugging in our values, we get:

y - 54 = 45(x - 4)

Simplifying, we get:

y - 54 = 45x - 180

y = 45x - 126

So the equation of the tangent line to the curve y = x^3 - 3x + 2 at the point (4, 54) is y = 45x - 126.

To find the equation of the tangent line to the curve y = x^3 - 3x + 2 at the point (4, 54), we need to first find the derivative of the function and then use the point-slope form of a line.

1. Find the derivative of the function with respect to x:
y'(x) = d/dx (x^3 - 3x + 2) = 3x^2 - 3

2. Evaluate the derivative at the given point (4, 54) to find the slope of the tangent line:
m = y'(4) = 3(4)^2 - 3 = 3(16) - 3 = 48

3. Use the point-slope form of a line (y - y1 = m(x - x1)):
y - 54 = 48(x - 4)

4. Simplify the equation:
y - 54 = 48x - 192
y = 48x - 138

So, the equation of the tangent line to the curve y = x^3 - 3x + 2 at the point (4, 54) is y = 48x - 138.

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The slope of the line is -3. The equation of the line is expressed as: y = -3x + 7.

The equation, y = mx + b, if the slope-intercept form of any straight line, where:

m is the slope = change in y / change in x = rise / run

b is the y-intercept or the point on the y-axis that the line cuts across.

From the graph, we know that:

b = 7 (y-intercept)

Slope (m) = rise/run = -3/1

Slope (m) = -3

Therefore, the equation of the line would be:

y = -3x + 7

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The number of ways we can pick five students for a student council if we have twelve people running would be 792 ways.

The formula needed to calculate the number of possible ways for a student council consisting of five individuals, from a pool of twelve candidates, is the combination equation.

n choose k = n! / (k! x (n - k)!)

= 12 ! / ( 5 ! ( 12 ! - 5 ) !)

= 12 ! / ( 5 ! x 7 !)

= 792 ways

Thus, it has been determined that there exist 792 feasible options for selecting members for this student council out of the aforementioned dozen contestants.

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If angle NPO and angle RPQ have the same measure is unknown without more information.

Given:

∠QPO= 140

a) As, from the figure

∠QPO + ∠NPO= 180  {linear Pair}

140 + ∠NPO = 180

∠NPO = 40 degree

b) ∠SPN+ ∠SPR + ∠RPQ = 180  {linear Pair}

90 + ∠SPR + 40 = 180

∠SPR = 180 - 130

∠SPR = 50 degree

c)If a ray PM is drawn dividing angle ∠SPN into two equal parts

Then each part is= 90/2 = 45 degree

d) No, ∠NPO and ∠RPQ do not have the same measure.

If angle NPO and angle RPQ both share rays with QPO, then they are both adjacent angles, which means they share a common vertex (point P) and a common side (ray PO).

However, we cannot conclude that they have the same measure based solely on this information.

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complete question:

The correct answer is d. 1, 3. Both paralleling and bite-wing techniques use the same principles of horizontal angulation.

Horizontal angulation refers to the angle at which the X-ray beam is directed from the X-ray tube towards the film or sensor, perpendicular to the long axis of the teeth. Correct horizontal angulation is crucial for obtaining accurate and clear radiographic images.
In the paralleling technique, the film or sensor is placed parallel to the long axis of the teeth, and the X-ray beam is directed perpendicular to both the teeth and the film/sensor. This technique reduces distortion and provides a more accurate representation of the teeth and surrounding structures.
The bite-wing technique is primarily used for interproximal (between teeth) radiography, capturing images of the crowns of the upper and lower teeth and the alveolar bone in a single exposure. The film or sensor is placed in the mouth with a tab or holder, and the patient bites down to stabilize it. The X-ray beam is directed at the same horizontal angle as in the paralleling technique, ensuring proper alignment with the teeth and film/sensor.
Although the bisecting technique also involves angulation, it uses a different principle known as the bisector angle. In this method, the film or sensor is positioned closer to the teeth, and the X-ray beam is directed at an angle between the long axis of the teeth and the film/sensor, creating a theoretical bisector. This technique may result in some image distortion compared to the paralleling and bite-wing techniques.

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The techniques Paralleling, Bisecting, and Bite-wing all use the principles of horizontal angulation in Dentistry. This angulation refers to the direction of the x-ray beam, which is adjusted in each of these methods to produce the optimal dental image.

The techniques that use the principles of horizontal angulation are Paralleling, Bisecting, and Bite-wing. This means the correct answer is 1, 2, 3 (option a). Horizontal angulation refers to the angle in which the beam of the x-ray is projected. In Dentistry, it signifies the movement of the x-ray tube head in a horizontal plane. For all three techniques, the x-ray beam's angulation is adjusted horizontally in order to attain the desired image.

Paralleling technique requires the dental film, tooth, and x-ray beam to be parallel to each other. Bisecting involves angulating the x-ray beam to correctly intercept the angle formed by the film and tooth. Bite-wing method also uses horizontal angulation by aligning the x-ray beam perpendicular to the curvature of the arch and the film.

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The parallelogram is a rhombus because all it's angle and sides are equal.

A rhombus is a special case of a parallelogram. In a rhombus, opposite sides are parallel and the opposite angles are equal. The properties of rhombus include;

1. All sides are equal in length.

2. Opposite angles are equal in a rhombus.

3. The diagonals bisect each other at 90 degrees.

4. Adjacent angles add up to 180 degrees.

Therefore since angle DCE = angle FCE

this means that angle FEC = DEC and thus angle D = angle F.

Therefore, we can say that the parallelogram is a rhombus because al it's sides and angles are equal.

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The width of the original piece of sheet metal is (w^2 - l^2)/(3w + 3l), and the length is (l^2 - w^2)/(3w + 3l).

To solve this problem, we can use the formula for the volume of a rectangular box, which is V = lwh, where l is the length, w is the width, and h is the height.

First, let's find the height of the box. Since we cut squares from each corner, the height of the box is the length of the square that was cut out. Let's call this length x.

The width of the box is the original width minus the lengths of the two squares that were cut out, which is w - 2x.

Similarly, the length of the box is the original length minus the lengths of the two squares that were cut out, which is l - 2x.

Now we can write the volume of the box in terms of x, w, and l:

V = (w - 2x)(l - 2x)(x)

Expanding this expression, we get:

V = x(4wl - 4wx - 4lx + 8x^2)

Simplifying further:

V = 4x^3 - 4wx^2 - 4lx^2 + 4wlx

To find the dimensions of the original piece of sheet metal, we need to maximize this volume. We can do this by taking the derivative of the volume with respect to x and setting it equal to zero:

dV/dx = 12x^2 - 8wx - 8lx + 4wl = 0

Solving for x, we get:

x = (2wl)/(3w + 3l)

Now we can use this value of x to find the width and length of the original piece of sheet metal:

w - 2x = w - 2(2wl)/(3w + 3l) = (w^2 - l^2)/(3w + 3l)

l - 2x = l - 2(2wl)/(3w + 3l) = (l^2 - w^2)/(3w + 3l)

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Answer:

The length of each edge of the cube is 21 cm.

Step-by-step explanation:

The surface area of a cube is given by the formula:

SA = 6s^2

where SA is the surface area and s is the length of each edge.

We can rearrange this formula to solve for s:

s = sqrt(SA/6)

Substituting the given value of SA = 2646 cm^2, we get:

s = sqrt(2646/6)

s = sqrt(441)

s = 21

Therefore, the length of each edge of the cube is 21 cm.

The area of the figure attached is 15 square meters

The figure is a parallelogram and area of a parallelogram is solved using the formula

= base x height

Information given in the problem includes:

base = 5 m

height = 3 m

plugging in the values in to the formula for area for area of a parallelogram will result to

= 5 m x 3 m

= 15 square meters

Hence the area of the parallelogram  attached is 15 square meters

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The modulus of rupture (MOR) can be calculated by dividing the maximum load by the section modulus. The section modulus for a rectangular cross-section is (width * height^2)/6.

In this case, the width is 2 inches and the height is 4 inches, so the section modulus is (2 * 4^2)/6 = 5.33 in^3. Therefore, the MOR = 240 kips / 5.33 in^3 = 45 kpsi.

The apparent modulus of elasticity (MOE) can be calculated using the formula MOE = (F * L^3) / (4 * h * d^3 * delta), where F is the maximum load, L is the span length, h is the height of the wood, d is the depth of the wood, and delta is the deflection at mid-span. Plugging in the values given, we get MOE = (240 kips * 4^3) / (4 * 4 * 2^3 * 2.4 in) = 1,333 kpsi. Therefore, the apparent MOE for the 2 * 4 wood lumber is 1,333 kpsi.

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The generalization about cedar trees is that the height of cedar trees varies from the mean by an average of 107.5

Average is the quotient obtained by dividing the sum total of a set of figures by the number of figures.

The IQR is the inter quartile range of a data. This means 50% of the data is the inter quartile range.

This means 50/100 × 492

= 246

The range is the difference between the largest and smallest

Range = 210 -16

= 194

Average = sum of height/ number of trees

= 210+16+40+130+49+200

= 107.5

therefore the average height of cedar tree is 107.5

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Answer: 924

Step-by-step explanation:

To generate 25 random variates from this distribution using VLOOKUP, you can follow these steps:

1. Create a table with two columns - "Cumulative Probability" and "Demand".
2. In the "Cumulative Probability" column, list the cumulative probabilities for each demand value. To do this, add up the probabilities for all demand values up to and including the current demand value. For example, for demand value 1, the cumulative probability would be 0.4 (0.2 + 0.2).
3. In the "Demand" column, list the demand values.
4. Use the VLOOKUP function to generate the random variates. For each variate, use the RAND() function to generate a random number between 0 and 1, and then use VLOOKUP to find the corresponding demand value based on the cumulative probability. For example, if the random number is 0.3, and the cumulative probability for demand value 2 is 0.7, then the VLOOKUP function would return a demand value of 2.

Here's an example of how the table and VLOOKUP formula would look:

| Cumulative Probability | Demand |
|-----------------------|--------|
| 0.4                   | 1      |
| 0.7                   | 2      |
| 1.0                   | 3      |
| 1.0                   | 4 or more |

Assuming the table is in cells A1:B4, the VLOOKUP formula for the first variate would be:

=VLOOKUP(RAND(), A1:B4, 2, TRUE)

This will generate a random variate from the distribution. Copy and paste the formula into 24 more cells to generate a total of 25 variates.

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The requried arc length of DF is given as 66°.

A circle with tangent and secant pair is shown, we have to determine the arc  DF.

The angle between the tangent and secant of the circle is given as:

∠E= DG - DF/2

Substitute the value in the above expression,
24 = 114 - x/2
48 = 114 - x
x = 114 - 48
x = 66°

Thus, the requried arc length of DF is given as 66°.

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Answer:

4. 8.25

Step-by-step explanation:

x/5.25=6.6/4.2

Cross Multiply
4.2x=34.65

Divide

8.25

Since the perimeter of the triangle is 50cm, the value of x is equal to 9 units.

In Mathematics and Geometry, the perimeter of a triangle can be calculated by using this mathematical equation:

P = a + b + c

Where:

P represents the perimeter of a triangle.

a, b, and c represents the side lengths of a triangle.

By substituting the given parameters or dimensions into the formula for the perimeter of a triangle, we have the following;

50  = x + 4 + 2x + 6 + x + 4

50 = 4x + 14

4x = 50 - 14

x = 36/4

x = 9 units.

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From the equation of the circle, the radius of the circle is 3 and the center lies at (1, 0)

A circle is a closed curve that is drawn from the fixed point called the center, in which all the points on the curve are having the same distance from the center point of the center. The equation of a circle with (h, k) center and r radius is given by:

(x-h)^2 + (y-k)^2 = r^2

This is the standard form of the equation. Thus, if we know the coordinates of the center of the circle and its radius as well, we can easily find its equation.

In this problem, we have an equation x^2 + y^2 - 2x - 8 == 0

The radius of this circle is 3 and the center is at (1, 0).

The correct option are A and C

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[A] This is likely because bedrooms, bathrooms, and the living area are all roughly measuring the same thing: the size of the property.

The most likely explanation for the negative coefficient for BEDROOMS is option A - a multicollinearity problem with the model. This means that the variables included in the model (bedrooms, bathrooms, and living area) are highly correlated with each other, making it difficult to distinguish their individual effects on the selling price. As a result, the coefficient for BEDROOMS may be affected and appear counterintuitive.

[A] This is likely because bedrooms, bathrooms, and living area are all roughly measuring the same thing: The size of the property.

The negative coefficient for bedrooms could be a result of multicollinearity, where the predictor variables (bedrooms, bathrooms, and living area) are highly correlated. In this case, they are all related to the property's size, making it difficult for the model to distinguish their individual effects on the selling price.

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You will need to make investments about $160.75 each month for the next year to reach a future price of $2000 with an 8% APR compounded monthly

To calculate the monthly investment required to attain a future cost of $2000 at an APR of 8% compounded monthly for a year, we will use the formulation for future value of an annuity:

[tex]FV = P * [(1 + r/n)^{(n*t)} - 1]/(r/n)[/tex]

Where:

Plugging in the values, we get:

[tex]2000 = P * [(1 + 0.08/12)^{(12*1)} - 1]/(0.08/12)[/tex]

solving for P, we get:

P = 160.75

Therefore, you will need to make investments about $160.75 each month for the next year to reach a future price of $2000.

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The most appropriate measure of variability for this data set is the IQR,

Option B is the correct answer.

Since,

An interquartile range is a measure of the difference between the upper and lower quartiles of a dataset.

We can find the values of the upper and lower quartile and median from a box plot.

The middle line is the median and the first line is the lower quartile and the last line is the upper quartile.

The formula used is Upper quartile - Lower quartile.

We have,

To determine the appropriate measure of variability for the given data, let's first calculate some descriptive statistics:

Mean: (2 + 10 + 10 + 12 + 12 + 20) / 6 = 11

Median: the middle value is 11, since there are an even number of values

Range: the difference between the maximum value (20) and the minimum value (2) is,

20 - 2 = 18

Interquartile range (IQR): Q3 (the third quartile) is the median of the upper half of the data, which is (12, 12, 20), and Q1 (the first quartile) is the median of the lower half of the data, which is (2, 10, 10). Thus, Q3 = 12 and Q1 = 10, and the IQR is 12 - 10 = 2.

Now, we need to determine which measure of variability is most appropriate for this data set.

Option A suggests using the range, which is the difference between the maximum and minimum values.

The range is affected by outliers and extreme values, and in this data set, the value of 19 is significantly larger than the other values, which might make the range an unreliable measure of variability.

Option B suggests using the IQR, which is a measure of the spread of the middle 50% of the data.

The IQR is less affected by extreme values than the range and is often preferred when there are outliers or skewed distributions.

Option C suggests using the mean, which is a measure of central tendency, not variability.

Option D suggests using the median, which is also a measure of central tendency, not variability.

Therefore,

The most appropriate measure of variability for this data set is the IQR,

Learn more about the interquartile range here:

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Complete question is,

The length of ribbons found at a seamstress are listed.

2, 10, 10, 12, 12, 20

What is the appropriate measure of variability for the data shown, and what is its value?

A. The range is the best measure of variability and equals 14.

B. The IQR is the best measure of variability and equals 2.

C. The mean is the best measure of variability and equals 11.

D. The median is the best measure of variability and equals 11.

The probability of getting a large or blueberry smoothie is 7/15.

The probability is calculated based on the following assumptions:

Probability of getting a large smoothie = 1/3

Probability of getting a blueberry smoothie = 1/5

P(large and blueberry) = P(large) * P(blueberry)

P(large and blueberry) = (1/3) * (1/5)

P(large and blueberry) = 1/15

P(large or blueberry) = P(large) + P(blueberry) - P(large and blueberry)

P(large or blueberry) = 1/3 + 1/5 - 1/15

P(large or blueberry)  = 7/15

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