Exercise 2.38. We choose one of the words in the following sentence uniformly at random and then choose one of the letters of that word, again uniformly at random: SOME DOGS ARE BROWN (a). Find the probability that the chosen letter is R. (b). Let X denote the length of the chosen word. Determine the probability mass function of X. (c). For each possible value k of X determine the conditional probability P(X k|X 3) Hint. The decomposition idea works just as well for conditional probabilities: if (B\, , Bn} is a partition of 2, then
n P(A| D) = ∑ P(ABk | D). k=1 (d). Determine the conditional probability P(the chosen letter is R | X > 3). (e). Given that the chosen letter is R, what is the probability that the chosen word was BROWN?

The requried properties are true for all the parallelograms are,
1. Opposite sides are parallel     2. Opposite sides are congruent

A parallelogram is a quadrilateral consisting of pairs of parallel sides.

The following properties are true for all parallelograms,
The opposite sides are parallel.
The Opposite sides are congruent.

All parallelograms have four congruent sides (property 1), four right angles (property 2), or exactly one pair of parallel sides (property 3). These properties only apply to specific types of parallelograms, such as rectangles or squares.

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The number of times the group need to fill the measuring spoon in order to perform the experiment is 12 times

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Let the amount of teaspoon of salt added to the solution be = ( 3/8 ) of spoon

Let the number of students = 4 students

The group only has ( 1/8 ) teaspoon measuring spoon

Substituting the values in the equation , we get

So , the amount of teaspoon of salt added to the solution by one student = amount of teaspoon of salt added to the solution / ( 1/8 )

On simplifying the equation , we get

The amount of teaspoon of salt added to the solution by one student = ( 3/8 ) / ( 1/8 )

The amount of teaspoon of salt added to the solution by one student = 3 times

And , the number of times the salt is added by 4 students = 4 x 3 = 12 times

Hence , the number of times is 12

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The coordinates of both positions of point B is (2√5,0) and B2: (0,2√5).

The intersection of the graph of the function with the y-axis gives y-intercept of that function. The y-intercept is the value of y on the y-axis at which the considered function intersects it.

Assume that we've got: y = f(x)

At y-axis, we've got x = 0, so putting it will give us the y-intercept.

Thus, y-intercept of y = f(x) is y = f(0)

We are given that;

Point b=  2 root 5 units

Now,

Since point B is 2√5 units away from point A, we can draw two circles with radius 2√5 centered at point A. The two intersections of these circles with the gridlines will give us the two possible positions of point B.

For example, if point A has coordinates (0,0) and the gridlines are the x-axis and y-axis, then the two possible positions of point B are:

B1: (2√5,0) - the intersection of the circle with the positive x-axis

B2: (0,2√5) - the intersection of the circle with the positive y-axis

If the gridlines are different, then the positions of point B will be different.

Therefore, by the intercept the answer will be (2√5,0) and B2: (0,2√5).

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Answer:

w = 12

Step-by-step explanation:

To simplify the equation, we have to isolate 'w'. To isolate 'w',

     [tex]\dfrac{w}{2}-21=-15\\\\\\\dfrac{w}{2 }-21+21=-15+21 \ \text{\bf (Addition property of equality)}\\[/tex]

                    [tex]\dfrac{w}{2}= 6[/tex]

                [tex]2*\dfrac{w}{2}=6*2 \ \text{\bf (Multiplication property of equality)}\\\\\\[/tex]

                     w = 12

The exact value of all six trigonometric functions:

sin(θ) = 0.55

cos(θ) = 0.83

tan(θ) = 0.66

cot(θ) = 1.51

sec(θ) = 1.2

cosec(θ) = 1.81

Let us assume that x represents the horizontal distance(x-axis), y represents the vertical distance and r be the radius.

here, r = 6 units and x = 5 units

Using Pythagoras theorem,

y = [tex]\sqrt{r^2-x^2}[/tex]

y =  [tex]\sqrt{6^2-5^2}[/tex]

y = 3.32 units

Now we find the exact value of all six trigonometric functions.

Let us assume that θ be an angle made by radius with positive x-axis.

sin(θ) = y/r

sin(θ) = 3.32 / 6

sin(θ) = 0.55

cos(θ) = x/r

cos(θ) = 5/6

cos(θ) = 0.83

tan(θ) = y/x

tan(θ) = 3.32 / 5

tan(θ) = 0.66

cot(θ) = x/y

cot(θ) = 5/3.32

cot(θ) = 1.51

sec(θ) = r/x

sec(θ) = 6/5

sec(θ) = 1.2

cosec(θ) = r/y

cosec(θ) = 6/3.32

cosec(θ) = 1.81

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a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.

In order to show mathematical issues, the "pie chart," often referred to as a "circle chart," divides the circular statistical visual into sectors or portions. A proportionate amount of the entire is indicated by each sector. The Pie-chart is now the most effective method for determining the composition of something. Pie charts typically take the role of other graphs, such as bar graphs, line plots, histograms, etc.

Given that, the average American spent $761.00.

a. Money spent on Food and Beverages is:

F = 761(20/100) = $152.2

b. Money spent on Entertainment is:

E = 761(13/100) = $98.93

c. The money spent on food is more than entertainment by:

A = 152.2 - 98.93 = $53.27

Hence, a. The average American spend on Food & Beverage is $152.2. b. The average American spend on Entertainment/Recreation is $98.93. c. They spent $53.27 more on food than on entertainment.

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The edge of the cube is equal to 2 m³.

There are different quadrilaterals, for example: square, rectangle, rhombus, trapezoid and parallelogram.  Each type is defined accordingly to its length of sides and angles. For example,  in a square, all angles are 90° and all sides present the same value.

A square is a 2D figure, when the square is associated with a 3D figure, the figure is called a cube.

The cube presents 6 faces with length (l), height (h) and width (w) equal. Thus, the cube presents congruent edges. The volume is given from the formula V=a³, where a is a congruent edge.

The exercise gives the volume of the cube 8m³. As presented previously, the edges of the cube are congruent and the formula by the volume is V=a³. Thus,

V=a³

8=a³

[tex]\sqrt[3]{a^3}[/tex]=[tex]\sqrt[3]{8}[/tex]

a=[tex]\sqrt[3]{2^3}[/tex]

a=2 m

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The equivalent ratio to the given expression 3mm : 1cm as required in the task content is; 3 : 10.

It follows from the task content that the equivalent ratio of 3mm : 1cm is to be determined .

Recall, 1cm is equivalent to 10 mm.

On this note, the given ratio can be written as; 3mm : 10 mm.

Consequently, the ratio which is equivalent to the given expression is; 3 : 10.

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The solution to the system of equations is x = -8, y = 9.

To solve the system of equations by elimination, we must manipulate the two equations so that when the equations are added or subtracted, one of the variables is eliminated.

One method is to multiply one or both equations by a constant, resulting in opposite coefficients for one of the variables. In this case, we can multiply the first equation by 4 and the second equation by -5, yielding the following y coefficients:

code :

(4)(5x + 4y = -4) => 20x + 16y = -16 (-5)

(4x + 4y = 4) => -20x - 20y = -20

We can now combine the two equations to eliminate y:

code :

20x + 16y = -16 \s-20x - 20y = -20 \s———————

0x - 4y = -36

When we simplify the result, we get:

code :

-4y = -36 y = 9

Now we can plug y = 9 back into one of the original equations to find x. Let's look at the first equation:

code :

5x+4y = -4 5x+4(9) = -4 5x+36 = -4 5x = -40 x = -8

As a result, the system of equations solution is x = -8, y = 9.

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The missing values in the table are

3/2, 4/3 and 4/9

The table shows a proportion of

oats =  flours * 4/9

the proportion was gotten from

1/3 / 3/4 = 4/9

The missing parts are

let the missing value be x

x * 4/9 = 2/3

x =  2/3 / 4/9

x = 3/2

3 * 4/9 = x

x = 4/3

1 * 4/9 = x

x = 4/9

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Since the optimal production level is within the company's maximum capacity of 150 items, the company should produce and sell 75 items to maximize profit.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.

To find the optimal production level a that maximizes profit, we need to determine the point at which marginal revenue equals marginal cost.

The marginal cost function is the derivative of the cost function, which is C'(a) = 50.

The marginal revenue function is the derivative of the revenue function, which is R'(a) = -0.5(3/8 - 110).

Setting these two equal and solving for a gives:

50 = -0.5(3/8 - 110)

a = 75

Since the optimal production level is within the company's maximum capacity of 150 items, the company should produce and sell 75 items to maximize profit.

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Answer: 17x = 54

x represents chau's savings before it increased

Answer:

$60  per TV sold

Step-by-step explanation:

Multiply $25 by 17 to find out how much she earned from the extended warranties

so 25 x 17 = 425

1445-425= 1020

now divide 1020 by 17 to find out how much neda earned from each TV

1020/17 = 60

The equation of the line is y = (2/3)x + 20/3.

The equation of a line is given by:

y = mx + c

where m is the slope of the line and c is the y-intercept.

Example:

The slope of the line y = 2x + 3 is 2.

The slope of a line that passes through (1, 2) and (2, 3) is 1.

We have,

The equation has:

Slope = 2/3

Passes through the line (-4, 4).

Now,

The equation of the line is y = mx + c.

m = 2/3

Consider (-4, 4) = (x, y)

So,

4 = (2/3)(-4) + c

4 = -8/3 + c

c = 4 + 8/3

c = (12 + 8) / 3

c = 20/3

Now,

y = (2/3)x + 20/3

Thus,

The equation of the line is y = (2/3)x + 20/3.

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The solution is, the product is 9.25* 10^83.2.

In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.

here, we have,

2.5\times 10^22.5×10^2 and 3.7 \times 10^53.7×10^5

=2.5* 10^22.5×10^2 × 3.7 * 10^53.7×10^5

=9.25* 10^24.5*10^58.7

=9.25* 10^83.2

Hence, The solution is, the product is 9.25* 10^83.2.

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Answer:

The slope is 3.

Step-by-step explanation:

You can find this by using rise/run.

The line goes from (1,2) to (2,5).

The rise for this is 3 and the run is 1.

3/1 is 3, therefore the slope is 3

The surface area of the composite figure is 415. 25 in²

From the image shown, we can see that the figure is made up of a cuboid and a semicircle.

The formula for calculating the surface area of a cuboid is given as;

Surface area =2lw+2lh+2hw

Given that, l is the length, h is the height and w, the width.

Surface area = 2(6×10) + 2(6×8) + 2(8 × 10)

expand the brakect

Surface area = 120 + 96+ 160

Surface area = 376 in²

Surface area of a semicircle = 1/2(πr2 ),

Substitute the values

Surface area = 1/2 ×22/7 ×5² = 39.25 in²

Hence, the value is the sum of the surface areas

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The areas of the triangles are as follows:

1. 52.5  ft²

2. 369 inches²

The area of a triangle can be represented as follows:

area of a triangle = 1 / 2 bh

where

Therefore, let's find the areas of the triangle using the formula as follows:

1.

area of the triangle = 1 / 2 × 15 × 7

area of the triangle = 1 / 2 × 105

area of the triangle =  52.5  ft²

2.

area of the triangle = 1 / 2 × 41 × 18

area of the triangle = 1 / 2 × 738

area of the triangle = 369 inches²

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The equation of the plane that passes through (-1, 3, 2) and is perpendicular to the planes x + 2y + 2z = 11 and 3x + 3y + 2z = 15 is -2x + 4y - 3z - 17 = 0

In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. A plane can be defined by an equation in the form of ax + by + cz = d, where a, b, and c are coefficients that determine the plane's orientation and d is a constant.

To find the equation of a plane, we need to know its normal vector, which is a vector perpendicular to the plane. We can find the normal vector of the plane we are looking for by taking the cross product of the normal vectors of the two given planes.

The normal vector of the first plane, x + 2y + 2z = 11, is (1, 2, 2), and the normal vector of the second plane, 3x + 3y + 2z = 15, is (3, 3, 2). To take their cross product, we can use the following formula:

n = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

where n is the normal vector, and a and b are the two given normal vectors. Plugging in the values, we get:

n = (2(2) - 2(3), 2(3) - 1(2), 1(3) - 2(3)) = (-2, 4, -3)

This means that the normal vector of the plane we are looking for is (-2, 4, -3). We also know that the plane passes through the point (-1, 3, 2), so we can use the point-normal form of the equation of a plane, which is:

a(x - x₀) + b(y - y₀) + c(z - z₀) = 0

where (x₀, y₀, z₀) is the given point, and a, b, and c are the coefficients of the normal vector. Plugging in the values, we get:

-2(x + 1) + 4(y - 3) - 3(z - 2) = 0

Simplifying, we get:

-2x + 4y - 3z - 17 = 0

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The measure of arc AD is 25°.

AC and BD are chords of the circle. The two chords intersect at the point E which makes an angle 93°. The measure of arc BC is 161°. To find the measure of arc AD:

The measure of arc AD by using the property that "if two chords intersect in the interior of the circle, then the measure of each angle is half the sum of the arcs intercepted by the angles and its vertical angle".

Thus, applying the above theorem:

93° = 1 /2 (arcBC + arcAD)

Here, measure of angle E is 93°.

93° = 1 / 2( 161 + AD)

93 = 161 / 2 + AD / 2

93 = 161/ 2 +  AD/2

solving further ( multiplying equation by 2 on both sides)

186 = AD + 161

AD = 186 - 161

arc AD = 25°

So, the measure of arc AD is  25°.

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____The given question is incomplete, the complete question is given below:

Find each value or measure. Assume that segments that appear to be tangent are tangent. Find arc AD.

The rate of return is an important measure of an investment's performance and is calculated by dividing the net investment income by the initial investment.

The formula for calculating a rate of return is expressed as the net investment income divided by the initial investment (ROI = (Gain from Investment – Cost of Investment) / Cost of Investment). To calculate the rate of return, you need to first determine the net investment income by subtracting the cost of investment from the gain from investment. Then, divide this number by the cost of investment and multiply it by 100 to convert it into a percentage. For example, if you invest $1000 and you make a total of 1200 back, your rate of return is 20%. This is calculated by subtracting the cost of investment (1000) from the gain (1200), resulting in a net income of 200. Then, divide $200 by the cost of investment (1000) to get 0.2 and multiply it by 100 to get a rate of return of 20%.

The rate of return is an important measure of an investment's performance and is calculated by dividing the net investment income by the initial investment.

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There was 19% percent error of the baker's prediction.

Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

Given that, a bakery sold 105 cupcakes in one day, the head baker predicted he would sell 85 cupcakes that day.

We are asked to find the percent error of the baker's prediction,

Percent error = |expected value - exact value| / exact value × 100 %

The expected value is the prediction of the Chef = 85

The exact value = 105

Percent error = |85-105| / 105 × 100 %

= 20/105 × 100 %

= 19%

Hence, there was 19% percent error of the baker's prediction.

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a) The pet shelter has 40% cats (20% black, 5% adopted), and 60% dogs (10% black, 10% adopted). 50% are adopted in non-black cats and dogs, b) 0.45, c) 0.19, d) 0.15, e) 0.375, f) 0.6.

a) The probability tree would seem like this:

P(Cat) = 40%

P(Black Cat) = 20%

P(Adopted Black Cat) = 5%

P(Not Adopted Black Cat) = 95%

P(Not Black Cat) = 80%

P(Adopted Not Black Cat) = 50%

P(Not Adopted Not Black Cat) = 50%

P(Dog) = 60%

P(Black Dog) = 10%

P(Adopted Black Dog) = 10%

P(Not Adopted Black Dog) = 90%

P(Not Black Dog) = 90%

P(Adopted Not Black Dog) = 50%

P(Not Adopted Not Black Dog) = 50%

b) P ( Adopted & Not Black Dog ) = P(Not Black Dog) x P(Adopted | Not Black Dog) = 0.90 * 0.50 = 0.45

c) P(Not Adopted & Black Cat) = P(Black Cat) x P(Not Adopted | Black Cat) = 0.20 * 0.95 = 0.19

d) P(Adopted) = P(Adopted Black Cat) + P(Adopted Not Black Cat) + P(Adopted Black Dog) + P(Adopted Not Black Dog) = 0.05 + 0.50 + 0.10 + 0.50 = 0.15

e) P(Adopted | Cat) = P(Adopted & Cat) / P(Cat) = (0.05 + 0.50) / 0.40 = 0.375

f) P(Cat | Adopted) = P(Cat & Adopted) / P(Adopted) = (0.05 + 0.50) / 0.15 = 0.6.

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The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.

here, we have,

The given values are

x,            f(x)

6,             -2

7,              4

8,              6

9,              4

10,            -2

The equation of the function in vertex form is given as follows;

f(x) = a × (x - h)² + k

To find the values of a, h, and k, we proceed as follows;

When x = 6, f(x) = -2

We have;

-2 = a × (6 - h)² + k = (h²-12·h+36)·a + k.............(1)

When x = 7, f(x) = 4

We have;

4 = a × ( 7- h)² + k = (h²-14·h+49)·a + k...........(2)

When x = 8, f(x) = 6...........(3)

We have;

6 = a × ( 8- h)² + k

When x = 9, f(x) = 4.

We have;

4 = a × ( 9- h)² + k ..........(4)

When x = 10, f(x) = -2...........(5)

We have;

-2 = a × ( 10- h)² + k

Subtract equation (1) from (2)

4-2 =  a × ( 7- h)² + k - (a × (6 - h)² + k ) = 13·a - 2·a·h........(6)

Subtract equation (4) from (2)

a × ( 9- h)² + k - a × ( 7- h)² + k

32a -4ah = 0

4h = 32

h = 32/4

  = 8

From equation (6) we have;

13·a - 2·a·8 = 6

-3a = 6

a = -2

From equation (1), we have;

-2 = -2 × ( 10- 8)² + k

-2 = -8 + k

k = 6

The equation of the function in vertex form is f(x) = -2·(x - 8)² + 6

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As a result, the answer is (e) 2.4 square inches, which is the difference between Richard's measured and real circle area.

In mathematics, area is a measure of the amount of space occupied by a two-dimensional object, such as a rectangle, triangle, circle, or any other shape. It's a scalar quantity that describes the size of a region in two-dimensional space. The units of area are typically square units, such as square inches, square centimeters, square meters, etc. In general, the area of a shape is a measure of how much space it occupies, and it is an important concept in geometry, engineering, and many other fields.

Here,

The formula for the area of a circle is given by:

A = πr²

Where r is the radius of the circle.

Using the incorrect radius of 3.6 inches, the calculated area would be:

A = π * (3.6 inches)² = 40.44 square inches

Using the correct radius of 3.5 inches, the actual area would be:

A = π * (3.5 inches)² = 38.5 square inches

So, the difference between Richard's measured area of the circle and the actual area of the circle would be:

40.44 square inches - 38.5 square inches = 1.94 square inches

Therefore, the answer is (e) 2.4 square inches that is the difference between Richard’s measured area of the circle and the actual area of the circle.

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Answer:

We can use the formula:

distance = rate x time

The total distance traveled by the plane is 3,000 + 600 = 3,600 miles. The time it took to cover this distance is 2 hours and 30 minutes, or 2.5 hours. So we have:

3,600 = rate x 2.5

Solving for the rate, we get:

rate = 3,600 / 2.5 = 1440 mph

Therefore, the answer is (A) 1440mph

1. To determine if the pair of planes 3x + 2y + z = 8 and x + 3y – z = 5 are parallel or intersecting, we can check if their normal vectors are parallel or not.

The normal vector of the first plane is <3, 2, 1>, and the normal vector of the second plane is <1, 3, -1>. These normal vectors are not parallel, so the planes intersect at a line.

To find a parametrization for the line of intersection, we can set one of the variables to be a parameter and solve for the other two. Let's choose z as the parameter.

From the second plane equation, we have z = x + 3y - 5. Substituting this into the first plane equation, we get: 3x + 2y + (x + 3y - 5) = 8

Simplifying, we get: 4x + 5y = 13

Solving for y in terms of x, we get: y = (13 - 4x) / 5

So a parametrization for the line of intersection is: x = t

y = (13 - 4t) / 5, z = t + 3((13 - 4t) / 5) - 5 = (2t + 2) / 5

2. To determine if the pair of planes 2c – y + z = 4 and 4c – 2y + 2z = 3 are parallel or intersecting, we can again check if their normal vectors are parallel or not.

The normal vector of the first plane is <2, -1, 1>, and the normal vector of the second plane is <4, -2, 2>. These normal vectors are parallel (in fact, they are scalar multiples of each other), so the planes are parallel.

To find the distance between the two parallel planes, we can take any point on one plane and find its distance to the other plane. Let's choose the point (0, 0, 4) on the first plane. The distance between this point and the second plane is: |4c - 2(0) + 2(4) - 3| / sqrt(4^2 + (-2)^2 + 2^2) = |4c - 5| / 2sqrt(6)

So the distance between the two parallel planes is |4c - 5| / 2sqrt(6).

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Answer:

  September, October, November

Step-by-step explanation:

When monthly rainfall in centimeters is represented by A(t) = 2.3sin(πt/6)+9.25 and A(0) represents July's rainfall, you want to know the months in which average rainfall is at least 10.5 cm.

We want to find the values of t that make A(t) ≥ 10.5:

  2.3sin(πt/6) +9.25 ≥ 10.5

  2.3sin(πt/6) ≥ 1.25 . . . . . . subtract 9.25

  sin(πt/6) ≥ 0.543478 . . . . divide by 2.3

  πt/6 ≥ 0.574575 . . . . . . . take inverse sine

  t ≥ 1.097

The sine function is symmetrical about π/2, so this also means solutions will be of the form ...

  πt/6 ≤ π -0.574575

  t ≤ 6 -1.097 ≈ 4.903

The month numbers that will have rainfall at least 10.5 cm will fall in the range ...

  1.097 ≤ t ≤ 4.903

  t ∈ {2, 3, 4}

If July is month 0, then these months are September, October, November.

Jacy's hometown will get at least 10.5 cm of rain in September, October, and November.

__

Additional comment

The attachment confirms this result. We have shifted the rainfall function so it can use conventional month numbers. It shows months 9, 10, 11 have rainfall above 10.5 cm.

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