Hw 17.1 (NEED HELPPP PLS)

Triangle proportionality, theorem

Answer:

The system of equations is:

-5x + 9y = -9

5x - 8y = 4

The determinant of the coefficients is:

D = |-5 9|

| 5 -8|

D = (-5)(-8) - (9)(5) = 40 - 45 = -5

The determinant of x is found by replacing the x-coefficients with the constants:

Dx = |-9 9|

| 4 -8|

Dx = (-9)(-8) - (9)(4) = 72 - 36 = 36

The determinant of y is found by replacing the y-coefficients with the constants:

Dy = |-5 -9|

| 5 4|

Dy = (-5)(4) - (-9)(5) = -20 + 45 = 25

Using Cramer's Rule:

x = Dx/D = 36/-5 = -7.2

y = Dy/D = 25/-5 = -5

Therefore, the solution to the system is x = -7.2 and y = -5.

Hope this helps!

The expression y - 1/x can be minimized to 0 but cannot be maximized for nonnegative values of x and y.

To maximize y-1/x, we want to find the largest possible value of y while keeping x as small as possible. Since x and y are nonnegative numbers and x*y=12, the smallest possible value of x is 0 and the largest possible value of y is infinity. Therefore, to maximize y-1/x, we need to set x=0 and y=∞, which gives us a value of infinity for y-1/x.

To minimize y-1/x, we want to find the smallest possible value of y while keeping x as large as possible. Since x and y are nonnegative numbers and x*y=12, the largest possible value of x is 12 and the smallest possible value of y is 1. Therefore, to minimize y-1/x, we need to set x=12 and y=1, which gives us a value of -1/12 for y-1/x.
To solve this problem, we need to maximize and minimize the expression y - 1/x given the constraint xy = 12 and x, y being nonnegative.

Step 1: Rewrite the constraint
Since xy = 12, we can write y in terms of x: y = 12/x

Step 2: Substitute y in the expression to be maximized/minimized
Now we can rewrite the expression as: y - 1/x = (12/x) - 1/x

Step 3: Simplify the expression
Combine the terms: (12 - 1)/x = 11/x

Step 4: Maximize and minimize the expression
Since x and y are nonnegative, we know x > 0 (otherwise, y would be undefined). As x approaches infinity, 11/x approaches 0, which means that the minimum value of the expression is 0.

However, there is no maximum value for the expression because as x approaches 0, 11/x approaches infinity.

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The expression y - 1/x can be minimized to 0 but cannot be maximized for nonnegative values of x and y.

To maximize y-1/x, we want to find the largest possible value of y while keeping x as small as possible. Since x and y are nonnegative numbers and x*y=12, the smallest possible value of x is 0 and the largest possible value of y is infinity. Therefore, to maximize y-1/x, we need to set x=0 and y=∞, which gives us a value of infinity for y-1/x.

To minimize y-1/x, we want to find the smallest possible value of y while keeping x as large as possible. Since x and y are nonnegative numbers and x*y=12, the largest possible value of x is 12 and the smallest possible value of y is 1. Therefore, to minimize y-1/x, we need to set x=12 and y=1, which gives us a value of -1/12 for y-1/x.
To solve this problem, we need to maximize and minimize the expression y - 1/x given the constraint xy = 12 and x, y being nonnegative.

Step 1: Rewrite the constraint
Since xy = 12, we can write y in terms of x: y = 12/x

Step 2: Substitute y in the expression to be maximized/minimized
Now we can rewrite the expression as: y - 1/x = (12/x) - 1/x

Step 3: Simplify the expression
Combine the terms: (12 - 1)/x = 11/x

Step 4: Maximize and minimize the expression
Since x and y are nonnegative, we know x > 0 (otherwise, y would be undefined). As x approaches infinity, 11/x approaches 0, which means that the minimum value of the expression is 0.

However, there is no maximum value for the expression because as x approaches 0, 11/x approaches infinity.

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Prisha has 56 apples and bananas. She has three times as many apples than bananas. Prisha has 42 apples.

To determine how many apples Prisha has, we will use the given information and set up an equation involving the terms apples and bananas.

Let A represent the number of apples and B represent the number of bananas.

According to the problem, A + B = 56.

It's also given that Prisha has three times as many apples as bananas, so A = 3B.

Now we can substitute the expression for A from Step 3 into the equation from Step 2:

3B + B = 56.

Combine the terms with B:

4B = 56.

Divide by 4 to find the value of B:

B = 14.

Now, using the value of B, find the value of A:

A = 3B = 3 × 14 = 42.

So, Prisha has 42 apples.

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The question asks you to analyze a statistical model used to estimate earnings based on various factors. You will need to interpret the significance of variables, coefficient estimates, and the overall fit of the model.

1. Without seeing the variables included in the earning equation, it is impossible to determine which variables are significant. However, the reported standard errors can be used to test the significance of each coefficient. If the absolute value of a coefficient divided by its standard error is greater than 1.96, then the variable is considered significant at the 5% level.

2. The coefficient estimate on females cannot be interpreted without seeing the entire equation.

3. R2 represents the proportion of variation in the dependent variable (in this case, earnings) that can be explained by the independent variables included in the equation. An R2 of 0.17 indicates that the included variables can explain 17% of the variation in earnings.

4. To test the joint significance of experience and females, we need to conduct an F-test.

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The two solutions for sin(x) = -0.81 on 0 ≤ x < 2π are a ≈ -2.207 and b ≈ 3.077, with a < b.

To solve sin(x) = -0.81 on 0 ≤ x < 2π, we first need to find the reference angle. We know that sin is negative in the third and fourth quadrants, so we need to find the angle whose sine is positive and then add π to get the angle in the third quadrant and subtract π to get the angle in the fourth quadrant.
Using a calculator, we can find that the reference angle for sin^-1(0.81) is approximately 0.935 radians or 53.5 degrees. To find the solutions in the third quadrant, we add π to the reference angle, giving us x = π + 0.935 ≈ 3.077 radians or x ≈ 176.5 degrees. To find the solutions in the fourth quadrant, we subtract π from the reference angle, giving us x = 0.935 - π ≈ -2.207 radians or x ≈ -126.5 degrees.
Therefore, the two solutions for sin(x) = -0.81 on 0 ≤ x < 2π are a ≈ -2.207 and b ≈ 3.077, with a < b.

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Answer:

-1

Step-by-step explanation:

3x^2-2x-5=0

(use the quadratic equation)

or

3(-1)^2-2(-1)-5=0

3+2-5=0

Okay, here are the steps to solve this problem:

* Maddy buys:

** 5 notebooks at $2.85 each = 5 * $2.85 = $14.25

** 3 pens at $1.79 each = 3 * $1.79 = $5.37

* Total cost = $14.25 + $5.37 = $19.62

* Maddy pays with $20

* Change = $20 - $19.62 = $0.38

Therefore, the change Maddy will receive is $0.38

Answer:

Maddy should get $0.38 or 38 cents back.

Step-by-step explanation:

First determine the cost of the 5 notebooks.

5 * 2.85 = 14.25

Then determine the cost of the 3 pens.

3 * 1.79=5.37

Add them together

14.25+5.37=19.62

Subtract the cost from 20 dollars.

20-19.62 = .38

Maddy should get 0.38 or 38 cents back.

Answer:

7

Step-by-step explanation:

Answer:

To find the unit rate of one aloo parate, we need to divide the total cost of 15 aloo parates by the number of aloo parates.

The cost per aloo parate can be calculated by dividing $75 by 15 as follows:

Cost per aloo parate = $75 ÷ 15 = $5

Therefore, the unit rate of one aloo parate is $5.

The area Covered by the sod is 118017.13Sq ft.

The Area covered by Dirt is 7049.6 Sq feet.

First, perimeter for Fencing

= ¼ x 2 x π x 380 + 2 x 15 +2 x 380 + ¼ x 2 x π x 15

= 197.5π + 190π

= 1410.5 feet.

Now, for Grass

= π/4 x (380 - 6)² + 87 ² - π/4 × (87 + 30)² + 2 x 380 x 15 + π/4 x 15² - (3/4) x π x 10² - 25π

= 31528π + 18969

= 118017.13

So, The area Covered by the sod is about 118017.13Sq ft.

Now, Area covered by Dirt

= π/4 x 380 ² - π/4 x (380 - 6)² + π/4 (87 + 30)² - 87² + π100

= (18613π - 30276)/4

= 7049.6 Sq feet

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The mayor should not be given an increase in staff based on the data provided, 5.66% of cities have a staff number of 9.

Based on the data provided, it appears that the majority of cities similar to Tums City have a staff size of either 8 or 7 employees. Only 12 out of the 212 cities surveyed have a staff size of 9, which is the same as the proposed increase.
The data does not provide a clear-cut answer, it does suggest that a staff size of 9 may not be necessary or common among similar cities. Additionally,

Budgetary concerns should not be overlooked, as the city council is responsible for ensuring responsible and sustainable use of resources.
Ultimately,

The decision to approve or deny the increase in staff size should be based on a thorough analysis of the city's specific needs, goals, and financial situation.
Size of Mayor's Staff | Number of Cities
5 | 21
6 | 39
7 | 31
8 | 98
9 | 12
10 | 11
The total number of cities in the data set.
Total number of cities = 21 + 39 + 31 + 98 + 12 + 11

Total number of cities = 212
The percentage of cities with each staff size.
5 staff: (21/212) * 100 = 9.91%
6 staff: (39/212) * 100 = 18.40%
7 staff: (31/212) * 100 = 14.62%
8 staff: (98/212) * 100 = 46.23%
9 staff: (12/212) * 100 = 5.66%
10 staff: (11/212) * 100 = 5.19%
The data, 46.23% of cities have a staff size of 8, which is the majority.

Only 5.66% of cities have a staff size of 9.

Given the budgetary concerns of the city council, it is reasonable to maintain the staff size at 8 employees since it is the most common among similar cities.

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The sampling distribution of the natural log of the odds ratio (lnOR) will be used for further statistical analyses to determine the relationship between smoking and breast cancer therefore lnOR  =0.86.

Based on the data provided from the case-control study of breast cancer and smoking, we can calculate the odds ratio (OR) to understand the association between smoking and breast cancer. Here's the data:

- Smoker: 25 cases, 15 controls
- Non-smoker: 75 cases, 85 controls

The odds ratio is calculated as (odds of exposure among cases) / (odds of exposure among controls), which is:

OR = (25/75) / (15/85) = 2.36

However, statistical inferences for odds ratios are based on the natural log of the odds ratio (lnOR) because the distribution for an odds ratio does not follow a normal distribution. To get the lnOR, we take the natural logarithm of the OR:

lnOR = ln(2.36) ≈ 0.86

The sampling distribution of the natural log of the odds ratio (lnOR) will be used for further statistical analyses to determine the relationship between smoking and breast cancer.

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Answer: random

Step-by-step explanation: i just did it

The point that maximizes the objective function is (9, 1) which gives the highest value of z = 53.

The answer is d. (9, 1).

To find the point that maximizes the objective function z = 6x - y, we need to evaluate the function at each given point and see which one gives the highest result.

a. z = 6(1) - 2 = 4
b. z = 6(1) - 5 = 1
c. z = 6(6) - 8 = 28
d. z = 6(9) - 1 = 53

To determine which point maximizes the objective function z = 6x - y, we will evaluate the function at each given point:

a. (1, 2): z = 6(1) - 2 = 4
b. (1, 5): z = 6(1) - 5 = 1
c. (6, 8): z = 6(6) - 8 = 28
d. (9, 1): z = 6(9) - 1 = 53

The point that maximizes the objective function is point d. (9, 1), with a value of z = 53.

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The inverses of these transformations and compositions include the following:

a. [tex]T_{(-5,1)}[/tex]

b. [tex]T_{(-2,-3)}R_{0,180^{\circ}}[/tex]

In Mathematics and Geometry, a transformation is the movement of a point from its initial position to a new location. This ultimately implies that, when a geometric figure or object is transformed, all of its points would also be transformed.

By critically observing the transformation rule and compositions, we can reasonably infer and logically deduce the following:

[tex]T_{(5,-1)}[/tex] = translation right 5 units and 1 unit down, so the inverse is left 5 units and 1 unit up i.e [tex]T_{(-5,1)}[/tex]

[tex]R_{0,180^{\circ}}T_{(2,3)}[/tex] = rotation of 180° about the origin and translation right 2 units and 3 unit up, so the inverse is [tex]T_{(-2,-3)}R_{0,180^{\circ}}[/tex]

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The point P(t) covers approximately 487.03 units

The parametric curve given by the equations x(t)=t2 13t−40 y(t)=t2 13t 1, to find the distance covered by the point P(t) = (x(t), y(t)) between t=0 and t=7,

we need to integrate the speed of the point over that time interval. The speed is given by the magnitude of the velocity vector:

|v(t)| = √[tex][x'(t)^2 + y'(t)^2][/tex]

where x'(t) and y'(t) are the derivatives of x(t) and y(t) with respect to t.

We can find the derivatives as follows:

x'(t) = [tex]2t(13t - 40) + t^{2(26)/ 3}[/tex]

y'(t) =[tex]2t(13t) + t^{2(13) / 3}[/tex]

Simplifying these expressions:

x'(t) = [tex]26t^{2 / 3} - 80t[/tex]

y'(t) =[tex]13t^{2 / 3} + 26t[/tex]

Therefore, the speed of the point is:

|v(t)| = √[tex][(26t^2 / 3 - 80t)^2 + (13t^2 / 3 + 26t)^2][/tex]

We can now integrate the speed over the interval t=0 to t=7:

distance = ∫(0 to 7) |v(t)| dt

This integral is difficult to solve by hand, but we can use numerical integration to get an approximate value.

Using a tool such as Wolfram Alpha or a numerical integration package in a programming language, we get:

distance ≈ 487.03

Therefore, the point P(t) covers approximately 487.03 units of distance between t=0 and t=7.

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The value of  f(-1) = 1 - (-1)²/₃ = 2/3 and f(1) = 1 - 1²/₃ = 2/3.

To find values c in (-1,1) such that f'(c) = 0, we take the derivative of f(x): f'(x) = -2x/3. Setting f'(c) = 0, we get -2c/3 = 0, which implies that c = 0. Therefore, the only value of c in (-1,1) such that f'(c) = 0 is c = 0.

Rolle's Theorem states that if a function is continuous on a closed interval [a,b], differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one point c in (a,b) such that f'(c) = 0.

In this case, f(x) satisfies the conditions of Rolle's Theorem on the interval [-1,1]. We have shown that there exists exactly one point c in (-1,1) such that f'(c) = 0, namely c = 0. Therefore, Rolle's Theorem holds true for f(x) on the interval [-1,1].

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(a)  This occurs when θ = 3π/2.

(b)  The range of the function is [0, 4].

(c)  This occurs when θ = 7π/6 and θ = 11π/6.

We have the function:

r = a + b cos(cθ) + p sin(qθ)

For the cardioid given by:

r = 2 + 2 sin θ

We can see that a = 2, b = p = 0, c = 1, and q = 1.

(a)  The cusp on the polar graph at the origin occurs when r = 0. Substituting the values of a, b, c, p, and q, we get:

0 = 2 + 2 sin θ

Solving for θ, we get:

sin θ = -1

This occurs when θ = 3π/2.

(b) The range of the function is the set of all possible values of r. From the given equation, we see that:

0 ≤ r ≤ 4

Therefore, the range of the function is [0, 4].

(c) For the function r = 1 + 2 sin θ, we can see that a = 1, b = p = 0, c = 1, and q = 2.

The inner loop on the polar graph occurs when r = 0. Substituting the values of a, b, c, p, and q, we get:

0 = 1 + 2 sin θ

Solving for θ, we get:

sin θ = -1/2

This occurs when θ = 7π/6 and θ = 11π/6.

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What is Vector field?

A vector field is a function that associates a vector to every point in a given space, commonly used in calculus and physics to model physical phenomena.

According toh the given information:

To sketch the vector field F(x, y) = ([tex]Y_{i} +X_{j}[/tex])/ sqrt(x² + y²), we can first analyze the behavior of the vector field at various points in the xy-plane.

Let's consider a few points:

1) At the origin (0,0), the denominator of the expression for F is undefined, so the vector field is not defined at this point.

2) Along the x-axis (y = 0), F(x,0) = xj / |x|, which means that the vectors point horizontally to the left for negative values of x and horizontally to the right for positive values of x.

3) Along the y-axis (x = 0), F(0,y) = yi / |y|, which means that the vectors point vertically upwards for positive values of y and vertically downwards for negative values of y.

4) At points away from the origin, we can analyze the direction of the vectors by considering the value of the expression [tex]Y_{i} +X_{j}[/tex] . If y is positive, then the vector will point upwards (in the positive y direction) and if y is negative, the vector will point downwards (in the negative y direction). Similarly, if x is positive, the vector will point towards the right (in the positive x direction) and if x is negative, the vector will point towards the left (in the negative x direction). The magnitude of the vectors decreases as we move away from the origin, because the denominator of the expression for F increases.

Based on this analysis, we can sketch the vector field.

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Using the vector end points the slope of the vector field is plotted.

The steps below are used to create a vector field:

a) Convert the supplied function to vector notation (also known as vector components form).

b) Define some arbitrary vector space points.

c) Apply the provided function to each of these points to determine the vector values.

d) Assess the arbitrary points as the absolute starting position and the arbitrary points plus vector values as the absolute finishing point.

Draw each of the aforementioned vectors such that it begins at the aforementioned starting point and finishes at the aforementioned computed finishing point.

For the given vector field we evaluate the function at different coordinates such as (0, 1), (0, -1), (1, 0), (-1, 0).

For f(0, 1) we have <1, 1>

For (1, 0) = <1, 1>

For (-1, 0) = <-1, -1>

Using the vector end points the slope of the vector field is plotted.

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The first part of the question asks us to complete the table and find the slope of the graph at the given point for two different functions. For the function (3x)^4, the derivative is 12(3x)^3. For the function r(θ) = 4 sin θ at the point (0,0), the slope of the graph is 4.

Let's complete the table and find the slope of the graph at the given point using the terms "derivative" and "slope."

1. Original Function: (3x)^4

Rewrite: (3x)^4
Differentiate: Using the power rule, d/dx[(3x)^4] = 4 * (3x)^(4-1) * d/dx(3x)
Simplify: 4 * (3x)^3 * 3 = 12(3x)^3

2. Function: r(θ) = 4 sin θ, Point: (0, 0)

To find the slope of the graph at the given point, we'll differentiate the function r(θ) with respect to θ.

Differentiate: dr/dθ = d/dθ [4 sin θ] = 4 * d/dθ [sin θ] = 4 * cos θ

Now, let's find the slope at the point (0, 0) by plugging θ = 0 into the derivative:

Slope: 4 * cos(0) = 4 * 1 = 4

So, the slope of the graph at the point (0, 0) is 4. To confirm your results, you can use the derivative feature of a graphing utility.

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The solution to this problem is given by: u(r) = u_ohi + (u_1 - u_ohi) * [(ln(r) - ln(a))/(ln(b) - ln(a))]. The temperature value at any point within the region between the two spheres, allowing you to understand the distribution of heat in this system.

The boundary value problem gives models the temperature distribution between two concentric spheres.

The solution of this problem can provide valuable information about the temperature at different points between the spheres.

The equation takes into account the distribution of temperature in the radial direction and the rate of change of temperature.

The values of u_ohi and u_1, which represent the temperature at the inner and outer sphere respectively, are important parameters in this problem.

The solution of this boundary value problem can be used to determine the temperature distribution in different spheres and to study heat transfer in various systems.

The boundary value problem you've provided, r d^2u/dr^2 + 2 du/dr = 0 with conditions u(a) = u_ohi and u(b) = u_1, models the temperature distribution between two concentric spheres of radii a and b, where a < b.

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The * values that make the solution a whole number are given as follows:

* = 4 or * = 9.

The divisibility rule for 5 states that a number is divisible by 5 if its ones digit (i.e., the digit in the units place) is either 0 or 5. In other words, if the number ends in 0 or 5, then it is divisible by 5.

The solution to the equation is obtained as follows:

5x = (516 + 49*)

x = (516 + 49*)/5

Hence it is needed one of these two following cases:

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The probability that the sequence of rolls is 2, 6, 1, 4, 5 is 1/7776, or as a decimal rounded to four decimal places, it is approximately 0.0001.

To find the probability that the sequence of rolls is 2, 6, 1, 4, 5 when a fair 6-sided die is rolled five times, we can use the following steps:
1. Since there are 6 sides on a fair die, the probability of rolling any specific number is 1/6.
2. The probability of rolling a specific sequence of numbers is the product of the probabilities of rolling each number in that sequence.
3. In this case, the sequence is 2, 6, 1, 4, 5. The probability of rolling each number in this sequence is 1/6.
So, to calculate the probability of the sequence 2, 6, 1, 4, 5, we multiply the probabilities of each individual roll:

(1/6) * (1/6) * (1/6) * (1/6) * (1/6) = 1/7776

Rounded to four decimal places, the probability is approximately 0.0001. Therefore, the probability that the sequence of rolls is 2, 6, 1, 4, 5 is 0.0001, or 1/7776 as a fraction.

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Answer:

  $669.30

Step-by-step explanation:

You want to know the amount Rapunzel saves each month if she saves 15% of her $4462 monthly income.

The amount is found by multiplying the income by the savings rate:

  15% × $4462 = 0.15 × $4462 = $669.30

Rapunzel saves $669.30 each month.

__

Additional comment

We can't tell if you're required to round this amount to the nearest dollar. If so, the rounded savings amount would be $669.

A calculator often omits trailing zeros. Its answer of 669.3 dollars will be expressed with 2 decimal places as 669.30, as monetary values usually are.

<95141404393>

Here, the data can be applied to understand the customer behavior, Stocked Pantry and preferences, adjust inventory, and optimize staffing and checkout procedures.

The Loaded Storage room supermarket places information about the quantity of things in stock by every client who goes through the express line.This information can be used in different ways to work on the store's tasks.

For example , the chief can use it to investigate client conduct and inclinations, distinguish well known things, and change stock likewise. The information can likewise be applied to enhance the store's staffing and checkout strategies.

In the event that the data shows that there is a top in express line traffic during specific times, the chief can plan gradually more staff during those times to guarantee speedy and productive help.

Generally, approaching this information can give significant experiences into the store's activities and assist the supervisor with pursuing informed choices that can further develop consumer loyalty and benefit.

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The complete question is

The manager at The Stocked Pantry grocery store can run a report to see the number of items purchased by each customer who goes through the express line. Customers in this line are allowed to purchase from 1 to 5 items. The table below shows the results from this morning.

Two different ways to write f dA as an iterated integral for the shaded region R. 1 + R 1, in the order dy dx and dx dy.

To write f dA as an iterated integral in two different ways for the shaded region R. 1 + R 1, we need to first determine the limits of integration for each variable.

If we start with the order dy dx, we can see that the shaded region is bounded by y = 0, y = 2, x = 1 and x = 2. Therefore, we can write the integral as follows:

f dA = ∫∫R f(x,y) dy dx
= ∫1^2 ∫0²-x f(x,y) dy dx + ∫2³ ∫0 f(x,y) dy dx
= ∫1^2 [∫0²-x f(x,y) dy] dx + ∫2³ [∫0 f(x,y) dy] dx
(Note: We split the integral into two parts based on the two different regions.)

Alternatively, if we switch the order to dx dy, we can see that the shaded region is bounded by x = 1, x = 2, y = x-1 and y = 2. Therefore, we can write the integral as follows:

f dA = ∫∫R f(x,y) dx dy
= ∫0 ∫x+1² f(x,y) dy dx + ∫1² ∫1 f(x,y) dx dy
= ∫0 [∫x+1² f(x,y) dy] dx + ∫1² [∫1 f(x,y) dx] dy
(Note: We split the integral into two parts based on the two different regions.)

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Two different ways to write f dA as an iterated integral for the shaded region R. 1 + R 1, in the order dy dx and dx dy.

To write f dA as an iterated integral in two different ways for the shaded region R. 1 + R 1, we need to first determine the limits of integration for each variable.

If we start with the order dy dx, we can see that the shaded region is bounded by y = 0, y = 2, x = 1 and x = 2. Therefore, we can write the integral as follows:

f dA = ∫∫R f(x,y) dy dx
= ∫1^2 ∫0²-x f(x,y) dy dx + ∫2³ ∫0 f(x,y) dy dx
= ∫1^2 [∫0²-x f(x,y) dy] dx + ∫2³ [∫0 f(x,y) dy] dx
(Note: We split the integral into two parts based on the two different regions.)

Alternatively, if we switch the order to dx dy, we can see that the shaded region is bounded by x = 1, x = 2, y = x-1 and y = 2. Therefore, we can write the integral as follows:

f dA = ∫∫R f(x,y) dx dy
= ∫0 ∫x+1² f(x,y) dy dx + ∫1² ∫1 f(x,y) dx dy
= ∫0 [∫x+1² f(x,y) dy] dx + ∫1² [∫1 f(x,y) dx] dy
(Note: We split the integral into two parts based on the two different regions.)

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The solution to the given boundary-value problem is y(x) = 2sin(pi*x) + x, where y(0) = 0 and y(1) = y'(1) = 0.

To solve the given boundary-value problem, we first write the differential equation in standard form

y'' + 7y = 7x

Next, we find the general solution of the homogeneous equation y'' + 7y = 0

The characteristic equation is r^2 + 7 = 0, which has roots r = ±√(7)i. Thus, the general solution of the homogeneous equation is

y_h(x) = c₁*cos(√(7)x) + c₂sin(√(7)*x)

where c₁ and c₂ are constants to be determined by the initial conditions.

Now, we find a particular solution of the non-homogeneous equation y'' + 7y = 7x

A particular solution of the non-homogeneous equation is y_p(x) = Ax + B, where A and B are constants. Substituting this into the differential equation, we get

0 + 7(Ax + B) = 7x

Solving for A and B, we get A = 1 and B = 0. Thus, a particular solution of the non-homogeneous equation is y_p(x) = x.

Therefore, the general solution of the given differential equation is

y(x) = c1*cos(√(7)x) + c2sin(√(7)*x) + x

Using the first initial condition y(0) = 0, we get

c1 = 0

Using the second initial condition y(1) = 0, y'(1) = 0, we get

c₂sin(√(7)) + 1 = 0

c₂sqrt(7)*cos(√(7)) = 0

Since c₂ cannot be zero, the second equation gives us cos(sqrt(7)) = 0, which implies sqrt(7) = (2n+1)*pi/2 for some integer n. Thus, the possible values of √(7) are

√(7) = pi/2, 3pi/2, 5pi/2, ...

Therefore, the general solution of the differential equation that satisfies the boundary conditions is

y(x) = (2/n)sin(npi*x) + x, where n is an odd integer.

In particular, the solution satisfying the given initial conditions is

y(x) = (2/1)sin(pix) + x = 2sin(pi*x) + x

Hence, the solution of the problem is y(x) = 2sin(pi*x) + x, where y(0) = 0 and y(1) = y'(1) = 0.

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Answer:

b) Dimensions of garden:
length = 10 feet
width = 3 feet

Step-by-step explanation:

I am only doing the part b. Part a is just a sketch of a rectangle with the dimensions computed

b) Find the dimensions of Amelia's garden

Let us use the variable L to represent the length and the variable W to represent the width of the garden

We are given that the length = 4 plus twice the width
In algebraic equation terms this would be
L = 4 + 2W

We are given that the area is 30 ft²
Area of a rectangle = LW
So
LW = 30

Substitute for L in terms of W:
(4 + 2W)W = 30
4W + 2W² = 30

Move 30 to the left and rearrange terms on the left
2W² + 4W - 30 = 0'

Divide by 2:
W² + 2W - 15 = 0

This is a quadratic equation that can be solved using factoring
Find the factors of -15 and see which of them when added will give a value of 2 which is the coefficient of W in the quadratic equation

Factors of -15 are
-15 1  => sum = 15 + 1 = - 14 X
-5 3  => sum = -5 + 3 = -2   X
5 -3  =>  sum = 5 + (-3) = 2  √

Given the correct factors we can rewrite the equation as
(W + 5)(W - 3) = 0

So either W + 5 = 0 or W - 3 =0

(If you multiply W + 5 by W - 3 you will get W² + 2W - 15)

Therefore the solutions to the quadratic equation are
W + 5 = 0   ==> W = -5  ; not possible, dimensions have to be positive
W - 3 = 0  ==> W = 3 ; this is the solution

So we have W, the width of the garden as 3 feet

Substitute W = 3 in the equation for length:
L = 4 + 2W

L = 4 + 2 x 3

L = 10

So length = 10 feet

To find vectors u→ and v→ in R3 such that w=span{u→,v→}, we can use the process of Gaussian elimination to solve the linear system of equations formed by equating each component of the vectors in w to the corresponding linear combination of the components of u→ and v→.

We start by setting up the following system of equations:

−a = xu + yv
−b = xv
−ab = yv

where x and y are scalar coefficients, and u = [1 0 0] and v = [0 1 0] are the standard basis vectors in R3.

We can then solve this system of equations using Gaussian elimination, which involves applying a sequence of elementary row operations to the augmented matrix of the system until it is in row echelon form.

The row echelon form of the augmented matrix is:

[ 1 0 a ]
[ 0 1 b ]
[ 0 0 0 ]

From this row echelon form, we can read off the solution as:

x = −b
y = ab
z = a

Thus, we have found that any vector w in the form [−a −b−ab] can be written as a linear combination of the vectors u→ = [1 0 −b] and v→ = [0 ab a], i.e., w = xu→ + yv→ for some scalars x and y.

Therefore, we have shown that w=span{u→,v→} with u→ = [1 0 −b] and v→ = [0 ab a].

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