A die is thrown twice. Let X1 and X2 denote the outcomes, and define random variable X to be the minimum of X1 and X2. Determine the distribution of X.

The points A(1, 1, 3), B(2, -5, 6), C(4, -2, -1), and D(3, 4, -4) form a parallelogram with an area of [tex]\sqrt{(1483)[/tex] square units.

To determine if the points A(1, 1, 3), B(2, -5, 6), C(4, -2, -1), and D(3, 4, -4) form the vertices of a parallelogram, we need to check two conditions:

Opposite sides are parallel.

Opposite sides are of equal length.

First, let's calculate the vectors representing the sides of the quadrilateral:

Vector AB = (2-1, -5-1, 6-3) = (1, -6, 3)

Vector BC = (4-2, -2-(-5), -1-6) = (2, 3, -7)

Vector CD = (3-4, 4-(-2), -4-(-1)) = (-1, 6, -3)

Vector DA = (1-3, 1-4, 3-(-4)) = (-2, -3, 7)

Next, we check if opposite sides are parallel. AB is parallel to CD, and BC is parallel to DA, as the direction ratios are proportional.

Now, we need to verify that opposite sides have the same length. Using the distance formula, we find:

[tex]|AB| = \sqrt{(1^2 + (-6)^2 + 3^2)} = \sqrt{(46)[/tex]

[tex]|CD| = \sqrt{((-1)^2 + 6^2 + (-3)^2)} = \sqrt{(46)[/tex]

[tex]|BC| = \sqrt{(2^2 + 3^2 + (-7)^2)} = \sqrt{(62)[/tex]

[tex]|DA| = \sqrt{((-2)^2 + (-3)^2 + 7^2)} = \sqrt{(62)[/tex]

Since opposite sides are both parallel and have equal lengths, the given points form the vertices of a parallelogram.

To find the area of the parallelogram, we can use the magnitude of the cross product of vectors AB and BC (or vectors BC and CD):

Area = |AB x BC| = |(1, -6, 3) x (2, 3, -7)| = |(-33, 17, 15)| = [tex]\sqrt{(33^2 + 17^2 + 15^2)} = \sqrt{(1483).[/tex]

So, the area of the parallelogram formed by the given points is sqrt(1483) square units.

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To find out if the given points A(1, 1, 3), B(2, −5, 6), C(4, −2, −1), and D(3, 4, −4) are the vertices of a parallelogram, we need to prove that the opposite sides are equal and parallel. In a parallelogram, both pairs of opposite sides are parallel and have the same length.

Step 1: Find the vectors representing the sides of the potential parallelogram.

Vectors AB, BC, CD, and DA can be calculated using the coordinates of points A, B, C, and D.

AB = B - A = (2 - 1, -5 - 1, 6 - 3) = (1, -6, 3)
BC = C - B = (4 - 2, -2 + 5, -1 - 6) = (2, 3, -7)
CD = D - C = (3 - 4, 4 + 2, -4 - (-1)) = (-1, 6, -3)
DA = A - D = (1 - 3, 1 - 4, 3 - (-4)) = (-2, -3, 7)

Step 2: Check if opposite sides are equal and parallel.

For sides AB and CD to be parallel and equal in length, vector AB should be equal to vector CD or AB should be a scalar multiple of CD.

Looking at vectors AB and CD:
AB = (1, -6, 3)
CD = (-1, 6, -3)

Vector CD is indeed the negative of vector AB, which means these vectors have the same length but opposite directions, so AB and CD are parallel and equal in length.

Now we need to check if vector BC is equal and parallel to vector DA.

BC = (2, 3, -7)
DA = (-2, -3, 7)

Vector DA is also the negative of vector BC. This means that they too have the same length but opposite directions, so BC and DA are parallel and equal in length.
Since both pairs of opposite sides are parallel and equal in length, points A, B, C, and D form a parallelogram.
Step 3: Calculate the area of the parallelogram.
The area of a parallelogram can be found using the cross product of two adjacent sides. The magnitude of the cross product vector gives us the area.
Let's calculate the cross product of vectors AB and AD:
AB = (1, -6, 3)
AD = (-2, -3, 7)  (Remember, AD is the opposite of DA)
The cross product AB x AD is:
| i   j  k  |
| 1  -6  3 |
| -2 -3  7 |
= i((-6)(7) - (3)(-3)) - j((1)(7) - (3)(-2)) + k((1)(-3) - (-6)(-2))
= i(-42 + 9) - j(7 + 6) + k(-3 - 12)
= i(-33) - j(13) + k(-15)
So the cross product of AB and AD is (-33, -13, -15).
The magnitude of this vector is √((-33)^2 + (-13)^2 + (-15)^2) = √(1089 + 169 + 225) = √(1483) ≈ 38.50 units squared.
Therefore, the area of the parallelogram formed by points A, B, C, and D is approximately 38.50 square units.

The decision rule in hypothesis testing is the criteria used to decide whether to accept or reject the null hypothesis. Given our level of significance and sample data, we reject the null hypothesis if the Z score (calculated as -1.92) is less than the critical value (-1.645). Hence, our decision rule is: if Z is less than -1.645, reject the null hypothesis.

The decision rule in hypothesis testing is the criteria that determines what the decision should be. In this case, the null hypothesis (H0) is that the mean weight of the chocolate chip bags is 439.0 grams, and the alternative hypothesis (H1) is that the mean weight is less than 439.0 grams because the machine is believed to be underfilling the bags.

Given a level of significance of 0.05 and a standard deviation of 21.0, we can calculate the z-score of the sample mean. Z = (Sample Mean - Population Mean) / (Standard Deviation / sqrt(Sample Size)), thus Z = (433.0 - 439.0) / (21.0 / sqrt(47)) = -1.92.

For a one-tailed test at a 0.05 level of significance, the critical value from the Z table is -1.645. The rule is to reject the null hypothesis if the calculated Z score is less than the critical value.

So the decision rule is: if Z is less than -1.645, reject the null hypothesis. In this case, as -1.92 is less than -1.645, we reject the null hypothesis. Hence, based on the sample, it can be said that the machine is underfilling the bags.

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Answer:

a number produced from a subset of the group of interest.

Step-by-step explanation:

This problems is, basically, about the difference of concepts between a statistic and a parameter.

When something is taken from a sample and estimated to the entire population, it is a statistic. For example, if you survey 200 Central New York residents, an 60% of them say they are Buffalo Bills fans, the 60% is a statistic.

When something is true to the entire population, it is a parameter. If i study the voting preference of all 53 players on the Buffalo Bills active roster, and 83% are Democrats, the 83% is an parameter.

So the correct answer is:

a number produced from a subset of the group of interest.

This subset is the sample, for example, and the group of interest is the population.

Answer:

L.A. = 80 + 16√13

Step-by-step explanation:

the lateral area is the area of the vertical faces.

So, for the given prism = The sum of the area of the vertical rectangles.

                                      = height * perimeter of the right triangle.

The hypotenuse of the right triangle = [tex]\sqrt{6^2+4^2} = \sqrt{36+16} =\sqrt{52} =\sqrt{4*13} =2\sqrt{13}[/tex]

So, the sides of the triangle are 4 , 6 and 2√13

The perimeter of the right triangle = 4 + 6 + 2√13 = 10 + 2√13

Height = 8

The lateral area for the prism = 8 * ( 10 + 2√13 ) = 80 + 16√13

Answer:

The correct answer is 80 + 16√13 feet²

Step-by-step explanation:

Like we can see in the plot, the prism has three rectangular sides, that are its lateral area. For calculating the area, we need to add up the three sides, this way:

Height of the prism (h) = 8 ' or 8 feet

Area of the first side = 8 * 4 = 32 feet²

Area of the third side = 8 * 6 = 48 feet²

Area of the third side = 8 * Hypotenuse of the triangle

Hypotenuse of the triangle² = 4² + 6² = 52

Hypotenuse of the triangle =√52 = √13 * 4 = 2√13 feet

Area of the third side = 8 * 2√13 feet = 16√13 feet²

Area lateral of the prism = 32 + 48 + 16√13 = 80 +  16√13 feet²

Answer:

∫(x+25x)dx=13x²

Step-by-step explanation:

From Exercise, we want  calculate integral for x + 25 x dx.  

But this not required  a substitution to express the integrand as a rational function. We can calculte integral, and we get

∫(x+25x)dx= ∫x dx+25∫x dx=x²/2+ \frac{25}{2} · x^2=\frac{26}{2}·x^2=13x²

Therefore, we get that is  

∫(x+25x)dx=13x².

Answer:

Step-by-step explanation:

Let

                               [tex]f(x,y,z) = z-x^2-y^4-e^{xy}[/tex].

The partial derivatives of this function are

                                    [tex]\frac{\partial f}{\partial x} (x,y,z) = ye^{xy} - 2x \\ \frac{\partial f}{\partial y} (x,y,z) = xe^{xy} - 4y^3 \\\frac{\partial f}{\partial z} (x,y,z) = 1[/tex]

The tangent plane equation through a point [tex]A(x_1,y_1,z_1)[/tex] is given by                     [tex]f'_x (x_1,y_1,z_1)(x-x_1) + f'_y(x_1,y_1,z_1)(y-y_1) + f'_z(x_1,y_1,z_1)(z-z_1) = 0[/tex]

In this case, we have

                                  [tex]x_1 = 1, y_1 = 0, z_1 = 2.[/tex]

The values of the partial derivatives in this point are

                                 [tex]\frac{\partial f}{\partial x} (1,0,2) = 0 \cdot e^{0} - 2\cdot 1 = -2 \\ \frac{\partial f}{\partial y} (1,0,2) = 1 \cdot e^{0} - 0 = 1 \\ \frac{\partial f}{\partial y} (1,0,2) = 1[/tex]

So, the equation is

                        [tex]-2(x-1) + 1 \cdot (y-0) + 1\cdot (z-2) = 0[/tex]

Therefore, the equation for the plane tangent to the surface at the point [tex](1,0,2)[/tex] is given by

                                      [tex]2x-y-z = 0[/tex]

Answer:

0.71%

Step-by-step explanation:

Given that a normally distributed set of data has a mean of 102 and a standard deviation of 20.

Let X be the random variable

Then X is N(102, 20)

We can convert this into standard Z score by

[tex]z=\frac{x-102}{20}[/tex]

We are to find the probability and after wards percentage of scores in the data set expected to be below a score of 151.

First let us find out probability using std normal table

P(X<151) = [tex]P(Z<\frac{151-102}{20} \\=P(Z<2.45)\\=0.5-0.4929\\\\=0.0071[/tex]

We can convert this into percent as muliplying by 100

percent of scores in the data set expected to be below a score of 151.

=0.71%

Answer:the constant of proportionality is 2.5 hours per day.

The equation is y = 2.5x

Step-by-step explanation:

Let x represent the number of days that Bruno listens to podcasts.

Let y represent the total number of hours that Bruno listens to podcasts.

Bruno listens to podcasts for

two and a half hours a day.

Let k represent the constant of proportionality. Therefore,

y = kx

k = y/x

So k = 2.5/1 = 2.5 hours/day

The equation is expressed as

y = 2.5x

By organizing the data and determining the minimum, maximum, median, and quartiles for the volumes of the largest dams in the United States and South America, we can construct boxplots to compare the distribution of the data in each region   Create a text based boxplot

United States:

 +---------+---------+---------+---------+---------+

 | Minimum |   Q1    | Median  |   Q3    | Maximum |

 +---------+---------+---------+---------+---------+

         |-----|-----|-----|-----|-----|

         50,000        77,700  77,854      108,814       125,628

South America:

 +---------+---------+---------+---------+---------+

 | Minimum |   Q1    | Median  |   Q3    | Maximum |

 +---------+---------+---------+---------+---------+

         |-----|-----|-----|-----|-----|

         46,563        51,403  103,979     292,783       311,539

To construct a boxplot for the volumes of the largest dams in the United States and South America, we will first need to organize the data in ascending order. Then we can determine the minimum, maximum, median, and quartiles. The boxplot will allow us to compare the distribution of the data for each region.

United States volumes (in cubic yards) in ascending order:

50,000

52,435

62,850

66,500

77,700

78,008

92,000

125,628

South America volumes (in cubic yards) in ascending order:

46,563

56,242

102,014

105,944

274,026

311,539

Now, we can find the minimum, maximum, median, and quartiles for each dataset:

United States:

- Minimum: 50,000

- Maximum: 125,628

- Median: The middle value is the average of the two middle values, (77,700 + 78,008) / 2 = 77,854

- First Quartile (Q1): The median of the lower half of the dataset, (52,435 + 62,850) / 2 = 57,642.5

- Third Quartile (Q3): The median of the upper half of the dataset, (92,000 + 125,628) / 2 = 108,814

South America:

- Minimum: 46,563

- Maximum: 311,539

- Median: The middle value is the average of the two middle values, (102,014 + 105,944) / 2 = 103,979

- First Quartile (Q1): The median of the lower half of the dataset, (46,563 + 56,242) / 2 = 51,402.5

- Third Quartile (Q3): The median of the upper half of the dataset, (274,026 + 311,539) / 2 = 292,782.5

With this information, we can construct boxplots for each region. The boxplots will visually represent the minimum, first quartile, median, third quartile, and maximum values for each dataset.

The population after one year would be 42 deer.

And, the population after two years would be 59 deer.

Used the concept of compound interest where the amount A after t years, starting from an initial value P and growing at a rate r is:  

[tex]A = P(1+r)^t[/tex]

Given that,

A population of 30 deer is introduced into a wildlife sanctuary.

Hence, After one year we have;

P₁ = 30 (1 + 0.40)¹  

= 42

After two years;

P₂ = 30 (1 + 0.40)²

= 59

Thus, the population after one year = 42 deer.

And, the population after two years = 59 deer.

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The possible values of net winnings in this coin flipping game range from -5 to 5 dollars. The individual probabilities of these outcomes can be calculated using a binomial probability distribution. The game is symmetric because of the fairness of the coin, resulting in the probability of winning and losing $5 being equal.

In this coin flipping game

, we have five independent trials (flips) each having two possible outcomes (heads or tails). This implies

a binomial distribution

is expected. In each trial, 'success' can be defined as getting heads, and 'failure' as getting tails. Number of successes (X) can range from 0 to 5, that is your net winnings can be from -5 to 5 dollars.

The Probability Mass Function (PMF) of X can be calculated using the binomial distribution formula, which is: P(X=k) = (5 choose k) * (0.5)^k * (0.5)^(5-k), where k represents number of 'successes' (heads), (5 choose k) is the number of combinations of getting k successful outcomes in 5 trials, and (0.5) is the probability of getting heads or tails.

PMF will give you the probability of each possible outcome (from -5 to 5 dollars), which in this specific scenario is symmetric due to the coin being fair. So for instance, winning $5 and losing $5 both have a probability of (0.5)^5 = 0.03125.

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Final answer:

To solve -30 ÷ (6a) when a = -5, substitute the value of a into the expression. The answer in simplest form is 1.

Explanation:

To solve the expression -30 ÷ (6a), we need to substitute the value of a, which is -5.

Substituting the value of a, -30 ÷ (6a) becomes -30 ÷ (6*(-5)).

Using the order of operations, first, we solve the multiplication inside the parentheses: -30 ÷ (-30).

Dividing a negative number by a negative number results in a positive number, so -30 ÷ (-30) = 1.

Therefore, the answer in simplest form is 1.

Answer:

D. Yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive.

Step-by-step explanation:

0.21 + 0.28 + 0.02 + 0.28 + 0.21 = 1

All individual data are between 0 and 1. Data added up = 1

The true statement is (d) Yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive.

For a distribution to be a discrete probability distribution, the following must be true

[tex]\sum P(x) =1[/tex]

So, we have:

[tex]\sum P(x) = 0.21 + 0.28 + 0.02 + 0.28 + 0.21[/tex]

Evaluate the sum

[tex]\sum P(x) =1[/tex]

The above statement shows that the distribution is a discrete probability distribution

Hence, the true statement is (d) Yes, because the probabilities sum to 1 and are all between 0 and 1, inclusive.

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Answer:

Mean of the 11th observations = 46.809

Step-by-step explanation:

The step by step explanations is given in the attached file.

What is appled is mean or average which is the total number of observation divided by the sum of the frequencies of each observaton .

Answer:

So the parallelogram is in the first and second quadrants.

Step-by-step explanation:

From Exercise we have that the parallelogram have a vetrices:

A(negative 1​,4​) ​B(4​,0​) ​C(12​,3​) ​D(7​,7​). We use a site geogebra.org to plof a graph for the given parallelogram.

From the graphs we can see that the parallelogram is mostly in the first quadrant and smaller in the second quadrant. So the parallelogram is in the first and second quadrants.

Answer:

The weight of concrete column is 1039 Newton.

Step-by-step explanation:

We are given the following in the question:

Diameter of column = 350 mm = 0.35 m

[tex]\text{Radius} = \dfrac{\text{Diameter}}{2} = \dfrac{0.35}{2} = 0.175 ~m[/tex]

Length of column = 2 m

Density of column = 2.45 Mg per meter cube

Volume of column =

[tex]\text{Volume of cylinder}\\= \pi r^2h\\\text{where r is the radius and h is the height}\\V = \dfrac{22}{7}\times (0.175)^2\times 2\\\\V = 0.1925\text{ cubic meter}[/tex]

Mass of column =

[tex]\text{Volume of column}\times \text{Density of cone}\\= 2.45\times 0.1925\\=0.4716~ Mg\\=0.4716\times 10^3~Kg\\= 471.6~Kg[/tex]

Weight of column =

[tex]\text{Mass}\times g\\= 471.6\times 9.8\\= 4621.68~ N[/tex]

Weight in pounds =

[tex]1 \text{ pound} = 4.4482 ~N\\\\\Rightarrow \dfrac{4621.68}{4.4482} = 1039\text{ pounds}[/tex]

The weight of concrete column is 1039 Newton.

Final answer:

To determine the weight of the concrete column in pounds, calculate its mass using the density formula and then convert it using the conversion factor. The weight of the column is approximately 1.882π lb.

Explanation:

To determine the weight of the concrete column in pounds, we need to calculate its mass and then convert it to pounds using the conversion factor. First, let's find the volume of the column by using the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.

The radius is half the diameter, so r = 350mm / 2 = 175mm = 0.175m. The height is given as 2m. Thus, the volume is: V = π(0.175m)^2 * 2m = 0.1925π m³.

Next, we can calculate the mass of the concrete using the density formula: mass = density * volume. Plugging in the given density of concrete (2.45 Mg/m³) and the calculated volume, we get: mass = 2.45 Mg/m³ * 0.1925π m³ = 0.425π Mg.

To convert the mass to pounds, we need to multiply by the conversion factor of 4.4482 N / 1 lb. Using this conversion factor, the weight of the concrete column is: weight = 0.425π Mg * 4.4482 N / 1 lb = 1.882π lb.

Answer:

[tex]\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy} = \boxed{\bold{4}}[/tex]

General Formulas and Concepts:
Calculus

Differentiation

Derivative Property [Multiplied Constant]:
[tex]\displaystyle \bold{(cu)' = cu'}[/tex]
Derivative Rule [Basic Power Rule]:

Integration

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \bold{\int\limits^b_a {f(x)} \, dx = F(b) - F(a)}[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \bold{\int {cf(x)} \, dx = c \int {f(x)} \, dx}[/tex]

Multivariable Calculus

Partial Derivatives

Vector Calculus

Circulation Density:
[tex]\displaystyle \bold{F = M \hat{\i} + N \hat{\j} \rightarrow \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}[/tex]

Green's Theorem [Circulation Curl/Tangential Form]:
[tex]\displaystyle \bold{\oint_C {F \cdot T} \, ds = \oint_C {M \, dx + N \, dy} = \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy}[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy}[/tex]

[See Graph Attachment] Points (0, 0) → (1, 0) → (1, 1) → (0, 1)

↓

[tex]\displaystyle \text{Region:} \left \{ {{0 \leq x \leq 1} \atop {0 \leq y \leq 1}} \right.[/tex]

Step 2: Integrate Pt. 1

Step 3: Integrate Pt. 2

We can evaluate the Green's Theorem double integral we found using basic integration techniques listed above:
[tex]\displaystyle \begin{aligned}\oint_C {y^2x \, dx + 9x^2y \, dy} & = \int\limits^1_0 \int\limits^1_0 {16xy} \, dx \, dy \\& = \int\limits^1_0 {8x^2y \bigg| \limits^{x = 1}_{x = 0}} \, dy \\& = \int\limits^1_0 {8y} \, dy \\& = 4y^2 \bigg| \limits^{y = 1}_{y = 0} \\& = \boxed{\bold{4}}\end{aligned}[/tex]

∴ we have evaluated the line integral using Green's Theorem.

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Topic: Multivariable Calculus

Unit: Green's Theorem and Surfaces

So if you want to fit the y-intercepts or "b", on the y-axis you should go by 25's [0, 25, 50, 75, 100...]

If the x-axis does not have to follow the same pattern (25's), you should go by 5's  [0, 5, 10, 15, 20...]

y = 7x + 50

y = 2x + 175

First I would plot the y-intercepts for each equation, then plot a few points with x = 5, 10, 15      Then draw a straight line.

The point where the two lines meet/cross paths is your solution. It should be (25, 225)  The x-axis is the number of miles, and the y-axis is the total cost. So Truck driver A and B would arrive/be at the same place/meet in 25 miles at the same cost of $225

Answer:

Option A is the correct answer.

Step-by-step explanation:

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis

change in the value of y = y2 - y1

Change in value of x = x2 -x1

y2 = final value of y

y 1 = initial value of y

x2 = final value of x

x1 = initial value of x

From the graph,

y2 = 6

y1 = 2

x2 = 3

x1 = 1

Slope,m = (6 - 2)/(3 - 1) = 4/2 = 2

To determine the intercept, we would substitute x = 3, y = 6 and

m= 2 into y = mx + c. It becomes

6 = 2 × 3 + c = 6 + c

c = 6 - 6 = 0

The equation becomes

y = 2x

Answer:

We calculate that is CD=4.

Step-by-step explanation:

From Exercise we have that

AB=AC=3

BC=6

BD=2

AD=1.  

From Axioms D-1 and D-3, we have  

D-1: ∀ A, B∈ S,  there is a unique AB ∈ R

D-3: ∀ A, B∈ S, AB=BA.

We use the  Axioms D-1 and D-3, and we get that

AC=CA=3

AD=1

because are the points is collinear, we get

CA+AD=CD

3+1=CD

CD=4

We calculate that is CD=4.

Final answer:

To find the length of segment CD, we assume points on a collinear path as A-D-B-C, and subtract the length of BD from BC, resulting in CD = 4 units.

Explanation:

The student has asked for the length of segment CD when given a set of collinear points A, B, C, and D, with various distances between them. Since we have point B between points A and C, and points A and D, and knowing AB, AC, BC, and BD, we can determine the length of AD by subtracting from AC the length of BC, since AB equals AC. Therefore, AD = AC - BC = 3 - 6 = -3, which does not make sense geometrically since lengths cannot be negative. This might imply a typo or a misinterpretation of the points' arrangement. If assuming points in a line segment: A-D-B-C, CD can be found by subtracting BD from BC, CD = BC - BD = 6 - 2 = 4.

Answer:

Step-by-step explanation:

a) C(x) = 6x+350

b) Function for total revenue, R(x) = 15x & charge per lawn is $15.

c) Jimmy must mow approx 39 lawns before he begins making a profit.

Consider the initial cost of lawnmower is $350,

Gasoline and maintenance cost are $6 per lawn.

(a) To formulate a function C(x) for the total revenue of mowing x lawn, we use the given values than the function C(x) will be given by

C(x) = 6x+350

Find the function for the total revenue from moving x lawns.

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

So, R(x) = P(x)+C(x)  ..............(1)

Substitute, P(x) = 9x-350 & C(x) = 6x+350 in (1) we get,

R(x) = 9x-350+6x+350

      = 15x

So, the total revenue = 15x

∴ Charge per lawn = R(x)/x = 15x/x = $15

To find the number of lawn jimmy must mow before begins making a profit, we use P(x) = 0

∴ 9x-350=0

⇒x = 350/9

⇒x ≈ 39

Hence, the solution is x = 39 (approx)

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Answer:

P(Sum of the two dice is 7) = 6/36

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have that:

A fair dice can have any value between 1 and 6 with equal probability. There are two fair dices, so we have the following possible outcomes.

Possible outcomes

(first rolling, second rolling)

(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)

(1,2), (2,2), (3,2), (4,2), (5,2), (6,2)

(1,3), (2,3), (3,3), (4,3), (5,3), (6,3)

(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)

(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)

(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)

There are 36 possible outcomes.

Desired outcomes

Sum is 7, so

(1,6), (6,1), (5,2), (2,5), (3,4), (4,3).

There are 6 desired outcomes, that is, the number of outcomes in which the sum of the two dice is 7.

Answer

P(Sum of the two dice is 7) = 6/36

Final answer:

The probability of getting a sum of 7 while rolling two fair dice is 1/6.

Explanation:

In the rolling of two fair dice, the probability of getting a sum of 7 is:

There are 6 ways to get a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

There are a total of 36 possible outcomes when rolling two dice, each with a probability of 1/36.

Therefore, the probability of rolling a sum of 7 is 6/36 or 1/6.

Answer:

Therefore, we conclude that the center of sphere at point

(27/3, -8/3, -8/3) with a radius 6.89.

Step-by-step explanation:

We have the formula for distance, we get

\sqrt{(x+3)^2+(y-4)^2+(z-4)^2} =2· \sqrt{(x-6)^2+(y-3)^2+(z+1)^2}

(x+3)^2+(y-4)^2+(z-4)^2=4·[(x-6)^2+(y-3)^2+(z+1)^2]

x²+6x+9+y²-8y+16+z²-8z+16=4x²-48x+4y²-24y+4z²+8z+184

3x²+3y²+3z²-54x+16y+16z=-143

(x²-54x/3)+(y²+16y/3)+(z²+16z/3)=-143/3

(x²-54x/3+729/9)+(y²+16y/3+64/9)+(z²+16z/3+64/9)=-143/3+729/9+2·64/9

(x-27/3)²+(y+8/3)²+(z+8/3)²=428/9

We calculate a radius \sqrt{428/9} =6.89

Therefore, we conclude that the center of sphere at point

(27/3, -8/3, -8/3) with a radius 6.89.

To find the equation of the sphere in question, use the property that the distance from P to A is twice the distance from P to B, and solve the resulting equation to find the sphere's center and radius.

To find an equation of a sphere where the distance from a generic point P to the point A(−3, 4, 4) is twice the distance from P to the point B(6, 3, −1), we use the geometric properties of spheres and distances in three-dimensional space. The distance from a point P to another point Q in 3D space, with coordinates P(x1, y1, z1) and Q(x2, y2, z2), can be found using the distance formula d = √((x2 − x1)² + (y2 − y1)² + (z2 − z1)²).

Let's denote the coordinates of P as (x, y, z). To satisfy the given condition, we have to solve the equation (x + 3)² + (y − 4)² + (z − 4)² = 4[(x − 6)² + (y − 3)² + (z + 1)²]. This equation is derived from setting the distance from P to A as twice the distance from P to B and then squaring both sides to eliminate the square root.

Solving this equation will give us the center and radius of the sphere.

Answer:

8(5) + 8(25) + 8(125) + 8(625) +8(3125)= 10(3125 - 1)=31240

Hence proved Conjecture is correct

L.H.S=R.H.S

Step-by-step explanation:

Consider the pattern:

8(5)=10(5-1)=40

8(5) + 8(25) = 10(25 - 1) =240

8(5) + 8(25) + 8(125) = 10(125 - 1) =1240

8(5) + 8(25) + 8(125) + 8(625) = 10(625 - 1)=6240

According to inductive reasoning next term of pattern will become:

8(5) + 8(25) + 8(125) + 8(625) +8(3125)= 10(3125 - 1)=31240

Checking whether conjecture is correct or not:

Consider L.H.S:

8(5) + 8(25) + 8(125) + 8(625) +8(3125)

40+200+1000+5000+25000

31240

R.H.S:

31240

Hence proved Conjecture is correct

L.H.S=R.H.S

Answer:

The value of x increased by 3.25.

Step-by-step explanation:

It is given that the value of x varies from x=2 to x=5.25.

We need to find the change is x.

If value of a variable x varies from a to b, then the change in te value of variable is

Change in x = b - a

For the given problem the variable is x, a=2 and b=5.25.

Change in x = 5.25 - 2

                  = 3.25

If the change is positive, then the value of variable increased and if the change is negative, then the value of variable decreased.

Therefore, the value of x increased by 3.25.

The value of x changed by 3.25 as it varies from x=2 to x=5.25.

The change in the value of x from x=2 to x=5.25 can be calculated by subtracting the initial value from the final value. In this case, the change in x is 5.25 - 2 = 3.25. Thus, the value of x has changed by 3.25.

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Answer:

[tex]z=\frac{0.427 -0.5}{\sqrt{\frac{0.5(1-0.5)}{110}}}=-1.531[/tex]  

[tex]p_v =P(z<-1.531)=0.0629[/tex]  

If we compare the p value obtained and using the significance level given [tex]\alpha=0.1[/tex] we have that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the proportion of tails is significantly less than 0.5.  

Step-by-step explanation:

1) Data given and notation

n=110 represent the random sample taken

X=47 represent the number of tails obtained

[tex]\hat p=\frac{47}{110}=0.427[/tex] estimated proportion of tails

[tex]p_o=0.5[/tex] is the value that we want to test

[tex]\alpha=0.1[/tex] represent the significance level

Confidence=90% or 0.90

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the proportion of tails is lower than 0.5:  

Null hypothesis:[tex]p\geq 0.5[/tex]  

Alternative hypothesis:[tex]p < 0.5[/tex]  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

3) Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.427 -0.5}{\sqrt{\frac{0.5(1-0.5)}{110}}}=-1.531[/tex]  

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided [tex]\alpha=0.1[/tex]. The next step would be calculate the p value for this test.  

Since is a left tailed test the p value would be:  

[tex]p_v =P(z<-1.531)=0.0629[/tex]  

If we compare the p value obtained and using the significance level given [tex]\alpha=0.1[/tex] we have that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the proportion of tails is significantly less than 0.5.  

Answer:

a) 25446.90 ft²

b) 101787.60 ft²

c) 407150.41 ft²

d) 2025πt²

Step-by-step explanation:

Data provided in the question:

Rate of spread radially = 45 feet per day

a) Radius of spread after 2 days

= 45 × 2

= 90 feet

Therefore,

Area affected = πr²

= π(90)²

= 25446.90 ft²

b) Radius of spread after 2 days

= 45 × 4

= 180 feet

Therefore,

Area affected = πr²

= π(180)²

= 101787.60 ft²

c) Radius of spread after 2 days

= 45 × 8

= 360 feet

Therefore,

Area affected = πr²

= π(360)²

= 407150.41 ft²

a) Radius of spread after t days

= 45 × t ft

Therefore,

Area affected = πr²

= π(45t)²

= 2025πt²

The area affected by the fungal disease, which spreads radially at a constant speed, can be calculated by using the formula for the area of a circle, where the radius is the product of the spread speed and time. After 2, 4, and 8 days, the areas affected are 25,446 square feet, 101,784 square feet, and 407,150 square feet, respectively.

The given scenario represents a real-world application of the mathematical concept involving growth in a circular pattern where the growth happens at a constant rate. Here, the representation of the disease spread through the farm is in form of a circle increasing in radius over time. To calculate the area affected on any given day (t), we apply the formula for the area of a circle (πr^2) where the radius is the rate of spread multiplied by the number of days.

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Answer:

16/52 or 4/13 when reduced.

Step-by-step explanation:

4/52 is for the Jack card since there are 4 jack cards out of a total of 52 cards.

12/52 is for the face cards since there are 12 face cards out of a total of 52 cards.

Since the problem says OR, this means one or the other, so you will be adding those two fractions together. If the problem said AND, this would mean that you have to multiply since both would have to occur.

The probability of selecting a heart or a face card from a standard deck of cards is 22/52.

To calculate the probability of selecting a heart or a face card from a standard deck of cards, we must consider the total number of heart cards and face cards in the deck, mindful not to double-count any cards that are both. A standard deck has 52 cards, 12 of which are face cards (3 face cards per suit, and there are 4 suits in total). Besides these 12 face cards, there are 13 heart cards, but since three of them are face cards, which we have already counted, we subtract those, leaving us with 10 additional heart cards. Thus, the total number of favorable outcomes is 12 (face cards) + 10 (non-face-card hearts) = 22.

The probability of drawing a heart or face card is then 22 favorable outcomes divided by 52 possible outcomes, which simplifies to 11/26 or about 42.31%.

To understand if this outcome is more or less likely than selecting a heart suit face card, we consider that there are only 3 face cards in the hearts suit. Hence, the probability of drawing a heart suit face card is 3/52 or about 5.77%. Clearly, drawing a heart or a face card is more likely than drawing a heart suit face card alone, as 42.31% > 5.77%.

Answer:

y(x)=C_1·e^{2x} + C_2·e^{-6x}

Step-by-step explanation:

From Exercise we have the differential equation

y''+4y'-12=0.  

This is a characteristic differential equation and we are solved as follows:

y''+4y'-12=0

m²+4m-12=0

m_{1,2}=\frac{-4±\sqrt{16+48}}{2}

m_{1,2}=\frac{-4±\sqrt{64}}{2}

m_{1,2}=\frac{-4±8}{2}

m_1=2

m_2=-6

The general solution of this differential equation is in the form

y(x)=C_1·e^{m_1 ·x} + C_2·e^{m_2 ·x}

Therefore, we get

y(x)=C_1·e^{2x} + C_2·e^{-6x}

The correct values of the constant [tex]\( r \)[/tex] for which [tex]\( y = e^{rx} \)[/tex] is a solution of the differential equation [tex]\( y'' + 4y' - 12y = 0 \)[/tex] are [tex]\( r = 2 \)[/tex] and [tex]\( r = -6 \).[/tex]

To find the values of [tex]\( r \)[/tex], we substitute [tex]\( y = e^{rx} \)[/tex] into the differential equation. First, we find the first and second derivatives of [tex]\( y \)[/tex]:

[tex]\[ y' = \frac{d}{dx}(e^{rx}) = re^{rx} \][/tex]

[tex]\[ y'' = \frac{d}{dx}(re^{rx}) = r^2e^{rx} \][/tex]

Now, we substitute [tex]\( y \)[/tex], [tex]\( y' \)[/tex], and [tex]\( y'' \)[/tex] into the differential equation:

[tex]\[ y'' + 4y' - 12y = 0 \][/tex]

[tex]\[ r^2e^{rx} + 4re^{rx} - 12e^{rx} = 0 \][/tex]

Since [tex]\( e^{rx} \)[/tex] is never zero, we can divide through by [tex]\( e^{rx} \)[/tex] to get the characteristic equation:

[tex]\[ r^2 + 4r - 12 = 0 \][/tex]

Now, we solve this quadratic equation for [tex]\( r \)[/tex]:

[tex]\[ r^2 + 6r - 2r - 12 = 0 \][/tex]

[tex]\[ r(r + 6) - 2(r + 6) = 0 \][/tex]

[tex]\[ (r - 2)(r + 6) = 0 \][/tex]

Setting each factor equal to zero gives us the possible values for [tex]\( r \)[/tex]:

[tex]\[ r - 2 = 0 \quad \Rightarrow \quad r = 2 \][/tex]

[tex]\[ r + 6 = 0 \quad \Rightarrow \quad r = -6 \][/tex]

Therefore, the values of [tex]\( r \)[/tex] for which [tex]\( y = e^{rx} \)[/tex] is a solution of the given differential equation are [tex]\( r = 2 \)[/tex] and [tex]\( r = -6 \).[/tex]

Answer:

a). [tex]4.86\times 10^{-15}[/tex] m

b). [tex]4.21\times 10^{-2}[/tex] m³

c). [tex]7.00\times 10^{3}[/tex] N²

Step-by-step explanation:

In this question we have to convert each option into SI units.

a). [tex]4.86(10^{-6})^{2}[/tex] mm

= [tex]4.86\times (10^{-12})\times (10^{-3} )[/tex] m

= [tex]4.86\times 10^{-15}[/tex] m

b). (348 mm)³

= [tex](348)^{3}\times (10^{-3})^{3}[/tex] m³

= [tex]42144192\times 10^{-9}[/tex] m³

= [tex]4.21\times 10^{-2}[/tex] m³

c). [tex](83700)^{2}\times (10^{-3})^{2}[/tex] N²

= [tex]700569\times 10^{4}\times 10^{-6}[/tex] N²

= [tex]7.00\times 10^{9}\times 10^{-6}N^{2}[/tex]

= [tex]7.00\times 10^{3}[/tex] N²