the average weight of 40 randomly selected minivans was 4,150 pounds. the minivan population standard deviation was 490 pounds. find the 99% confidence interval of the true mean weight of minivans.

The maximum length of each prism is equal to 7.99 centimeter.

Maximum surface area = 275 square centimeter (cm²).

Given, Light fixtures of regular hexagonal prism .

Determine the maximum surface area of this regular hexagonal prism by using this mathematical expression:

Maximum surface area (quantity) = Cost/unit price

Maximum surface area (quantity) = $11/$0.04

Maximum surface area (quantity) = 275 square centimeter (cm²).

Mathematically, the surface area of a regular hexagonal prism can be calculated by using this formula:

[tex]A = 6al + 3\sqrt{3} a^2[/tex]

Where:

A represents the surface area of a regular hexagonal prism.

a represents the edge length (apothem) of a regular hexagonal prism.

l represents the length of a regular hexagonal prism.

Substituting the given parameters into the formula, we have;

[tex]275 = 6 \times 4l + (3\sqrt{3} \times 4^2)[/tex]

[tex]275 = 24l + 48\sqrt{3} \\24l = 275 - 48\sqrt{3}\\ 24l = 191.8616\\l = 191.8616/24[/tex]

Length, l = 7.99 centimeter.

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Roy should cut the pizza at an angle of 4.7 radians if the total diameter of the pizza is 12 inches.

Diameter of the pizza = 12 inches

The area he can eat = 84. 8 inches

The area of the pizza for 12 inches diameter can be calculated by using the formula:

Area = π*[tex]r^2[/tex]

Area = π* ([tex]6^2[/tex])

Area = 36π

The angle of the slice of pizza he can eat about 84.8 square inches can be calculated as:

Area of sector = (θ/2) ×[tex]r^2[/tex]

84.8 = (θ/2) × [tex]6^2[/tex]

84.8 = 18*θ

θ = 84.8 / 18

θ = 4.71 radians

Therefore, we can conclude that Roy should cut the pizza at an angle of  4.7 radians.

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In Roman Numerals, smaller numbers can be subtracted from larger numbers, but only to a certain extent.

The number VCX is incorrect in Roman Numerals because V (5) cannot be subtracted from X (10) to make 5, and C (100) cannot be subtracted from X (10) to make 90.

In Roman Numerals, smaller numbers can be subtracted from larger numbers, but only to a certain extent.

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Answer:

Step-by-step explanation

Domain :

x

>

4

, in interval notation :

(

4

,

∞

)

Range:

g

(

x

)

∈

R

, in interval notation :

(

−

∞

,

∞

)

Explanation:

g

(

x

)

=

ln

(

x

−

4

)

;

(

x

−

4

)

>

0

or

x

>

4

Domain :

x

>

4

, in interval notation :

(

4

,

∞

)

Range: Output may be any real number.

Range:

g

(

x

)

∈

R

, in interval notation :

(

−

∞

,

∞

)

graph{ln(x-4) [-20, 20, -10, 10]} [Ans]  x>4

Answer:

Step-by-step explanation:

The Domain of g(x) = -|x| is all real numbers (no restrictions on what values x can take).

The Range of g(x) = -|x| is all real numbers less than or equal to zero. Absolute value of any real number is always greater than or equal to zero, and multiplying by a negative sign, that flips the sign of the result. So, g(x) will always be less than or equal to zero.

Domain:  (-∞, ∞), {x|x ∈ R}

Range: (-∞, 0), {y ≤ 0}

We can conclude that Jack's claim is correct. QRST is indeed a parallelogram, since opposite angles in QRST are congruent, and opposite sides in a parallelogram are also congruent

To determine whether Jack's claim that QRST is a parallelogram is correct, we need to use the properties of parallelograms. One of the properties of a parallelogram is that opposite angles are congruent. Therefore, we need to check if the opposite angles in QRST are congruent.

If m∠R = 72 and m∠T = 108, then the sum of these angles is 180 degrees (72 + 108 = 180). This indicates that angles R and T are supplementary.

If m∠S = 72, then we need to find the measure of angle Q. Since QRST is a quadrilateral, the sum of its interior angles is 360 degrees.

m∠Q + m∠R + m∠S + m∠T = 360

Substituting the given values, we get:

m∠Q + 72 + 72 + 108 = 360

Simplifying the equation, we get:

m∠Q = 108

Therefore, angles Q and S are congruent (both measuring 72 degrees) and angles R and T are supplementary (measuring 72 and 108 degrees, respectively). Since opposite angles in QRST are congruent, and opposite sides in a parallelogram are also congruent, we can conclude that Jack's claim is correct. QRST is indeed a parallelogram.

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The equation of the axis of symmetry for the given function is x = 1.

To find the equation of the axis of symmetry for the function f(x) = -4x² + 8x - 28, we can use the formula:

x = -b / (2a)

where "a" and "b" are coefficients in the quadratic equation ax² + bx + c.

In this case, a = -4 and b = 8. Plugging these values into the formula, we get:

x = -8 / (2*(-4))

x = -8 / (-8)

x = 1.

The axis of symmetry for a quadratic function in the form of [tex]f(x) = ax^2 + bx + c[/tex]  can be found using the formula x = -b / (2a).

In the case of the given quadratic function f(x) = -4x² + 8x - 28, the coefficient of [tex]x^2[/tex] is a = -4 and the coefficient of x is b = 8.

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Answer: 7x=30.5

Step-by-step explanation: If you were to answer this equation with the given information it would not be correct. 7(4.25)=29.75.  You need to go through BEDMAS to answer this.

You what you need to do is divide 30.5 by 7 to get the amount she needs to jog for a week, if you do that you get 4.37 miles each day to get to 30.5 in a week.

Corresponding angles have the same position on the parallel lines, alternate interior angles are inside and opposite, alternate exterior angles are outside and opposite, and consecutive interior angles are on the same side.

When two parallel lines are intersected by a transversal, there are several types of angle pairs that are formed. Corresponding angles are pairs of angles that are located in the same position on the parallel lines relative to the transversal. They have the same measure and are congruent.

Alternate interior angles are pairs of angles that are located on opposite sides of the transversal and inside the parallel lines. They are congruent and have the same measure. Alternate exterior angles are pairs of angles that are located on opposite sides of the transversal and outside the parallel lines. They are congruent and have the same measure.

Consecutive interior angles are pairs of angles that are located on the same side of the transversal and inside the parallel lines. They add up to 180 degrees.

To classify each angle pair, we need to determine their positions relative to the parallel lines and the transversal. By knowing the classifications, we can identify each angle pair and their properties.

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The results of the survey suggest that the student's claim of three-quarters of 17-year-olds having their driver's licenses may be accurate, but further research would be necessary to confirm this with a larger and more representative sample.

Based on the computer output, there is no convincing evidence that the true proportion of 17-year-olds at this high school with driver's licenses is not 0.75. This is because the P-value of 0.9315 is very large, indicating that the results of the survey are not statistically significant. Additionally, the 95% confidence interval for the sample proportion of 0.755556 includes 0.75, further supporting the claim that the true proportion may be close to 0.75.

It is important to note that the correct significance test was performed, testing the null hypothesis that p = 0.75 against the alternative hypothesis that p ≠ 0.75. This is the appropriate test when the claim being tested is about a specific value of the proportion, as in this case. The alternative hypothesis being p > 0.75 or p < 0.75 would be incorrect, as it assumes a one-sided test rather than a two-sided test.

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The area of a sector with a central angle of 45° and a diameter of 5.6 in. is 1.23 square inches.

To see why, you can use the formula for the area of a sector, which is:

A = (θ/360) x π x r^2

where θ is the central angle in degrees, r is the radius, and π is approximately 3.14.

First, you need to find the radius of the sector, which is half of the diameter:

r = d/2 = 5.6/2 = 2.8 in.

Next, you can plug in the values for θ and r into the formula:

A = (45/360) x 3.14 x 2.8^2 = 1.23 square inches

Therefore, the area of the sector is 1.23 square inches.

The area of a sector with a central angle of 120° and a radius of 18.4 m is 1908.57 square meters.

To see why, you can use the same formula for the area of a sector:

A = (θ/360) x π x r^2

First, you need to convert the radius from meters to centimeters, since π is in terms of centimeters:

r = 18.4 m x 100 cm/m = 1840 cm

Next, you can plug in the values for θ and r into the formula:

A = (120/360) x 3.14 x 1840^2 = 1908.57 square meters

Therefore, the area of the sector is 1908.57 square meters.

To find the length of the opposite side and the adjacent side, we can use the ratios of the sides in a 30-60-90 degree triangle.

The ratio of the opposite side to the hypotenuse is 1:2, and the ratio of the adjacent side to the hypotenuse is √3:2.

Using these ratios, we can find the length of the opposite side and the adjacent side as follows:

Opposite side = 1/2 x hypotenuse = 1/2 x 10 = 5 units

Adjacent side = √3/2 x hypotenuse = √3/2 x 10 = 5√3 units

Given a right triangle with an acute angle of 60 degrees and an adjacent side of 5 units, find the length of the hypotenuse and the opposite side.

To find the length of the hypotenuse and the opposite side, we can use the ratios of the sides in a 30-60-90 degree triangle.

The ratio of the hypotenuse to the adjacent side is 2:1, and the ratio of the opposite side to the adjacent side is √3:1.

Using these ratios, we can find the length of the hypotenuse and the opposite side as follows:

Hypotenuse = 2 x adjacent side = 2 x 5 = 10 units

Opposite side = √3 x adjacent side = √3 x 5 = 5√3 units

Given a right triangle with an acute angle of 45 degrees and an opposite side of 7 units, find the length of the hypotenuse and the adjacent side.

To find the length of the hypotenuse and the adjacent side, we can use the ratios of the sides in a 45-45-90 degree triangle.

In this type of triangle, the opposite side and the adjacent side are equal, and the hypotenuse is √2 times the length of the legs.

Using these ratios, we can find the length of the hypotenuse and the adjacent side as follows:

Opposite side = Adjacent side = 7 units

Hypotenuse = √2 x opposite side = √2 x 7 = 7√2 units

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Optimize crate cost by expressing material cost in terms of x and y, calculating cost of all four sides, and finding minimum cost. Optimal dimensions: x = 16 ft, y = 48 ft. Minimum cost: $8704.

To find the dimensions of the crate that will minimize the total cost of material, we need to use optimization techniques, we need to express the cost of materials in terms of x and y then calculate the cost of all four sides, then find the minimum cost.
Let's start by defining the variables we need to work with:

Let x be the length of one side of the square base (in feet),  Let y be the height of the crate (in feet).
From the given volume, we know that:
V = x^2 * y = 12288 ft^3
We can use this equation to solve for one of the variables in terms of the other:
y = 12288 / (x^2)
Now we need to express the cost of materials in terms of x and y.
The area of the bottom is x^2, so the cost of the bottom is:
[tex]C_b = 10 * x^2[/tex]
The area of each side is x * y, and there are four sides, so the cost of the sides is:
[tex]C_s = 4 * 2 * x * y = 8xy[/tex]
The area of the top is also x^2, so the cost of the top is:
[tex]C_t = 2 * x^2[/tex]
The total cost of materials is the sum of these three costs:
[tex]C = C_b + C_s + C_t = 10x^2 + 8xy + 2x^2[/tex]
Now we can substitute y = 12288 / (x^2) into this equation:
[tex]C = 10x^2 + 8x * (12288 / x^2) + 2x^2[/tex]
Simplifying this expression, we get:
[tex]C = 12x^2 + 98304 / x[/tex]
To find the minimum cost, we need to find the value of x that minimizes this expression. We can do this by taking the derivative of C with respect to x and setting it equal to zero:
C' = 24x - 98304 / x^2 = 0
Solving for x, we get:
x = 16 ft
Now we can use this value of x to find y:
y = 12288 / (16^2) = 48 ft
Therefore, the dimensions of the crate that will minimize the total cost of material are:
- Length of one side of the square base = x = 16 ft
- Height of the crate = y = 48 ft
To check that this is indeed the minimum cost, we can plug these values back into the expression for C and calculate the cost:
C = 10 * 16^2 + 8 * 16 * 48 + 2 * 16^2 = 8704
Therefore, the minimum cost of material for the crate is $8704.

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a. Null hypothesis (H0): μ1 = μ2 and Alternative hypothesis (H1): μ1 < μ2. b. The test statistic is -2.57 and p-value is  0.005 for the hypothesis test. c. At α = 0.05 it can be concluded that CPAs in states with flat state income tax rates work fewer hours per week during tax season compared to the average U.S. CPAs.

a. First, let's formulate the hypotheses:

Null hypothesis (H0): μ1 = μ2, which means that the mean hours worked per week during tax season by CPAs in states with flat state income tax rates is equal to the mean hours worked per week by all U.S. CPAs during tax season.

Alternative hypothesis (H1): μ1 < μ2, which means that the mean hours worked per week during tax season by CPAs in states with flat state income tax rates is less than the mean hours worked per week by all U.S. CPAs during tax season.

b. Now, let's compute the test statistic and p-value using the given sample data:

Sample mean (x) = 55 hours

Population mean (μ) = 60 hours

Population standard deviation (σ) = 27.4 hours

Sample size (n) = 150

We'll use the z-test for this hypothesis test:

z = (x - μ) / (σ / √n) = (55 - 60) / (27.4 / √150) ≈ -2.57

To find the p-value, we need to look up the z-value in the standard normal table, which gives us a p-value of approximately 0.005.

c. Lastly, let's draw our conclusion using α = 0.05:

Since the p-value (0.005) is less than α (0.05), we reject the null hypothesis (H0). This suggests that CPAs in states with flat state income tax rates work fewer hours per week during tax season compared to the average U.S. CPAs.

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Answer:

They can make 5 complete teams of 8 children even after one child goes home.

Step-by-step explanation:

If there are 43 children and they want to make teams of 8, we can find out how many complete teams they can make by dividing the total number of children by the number of children per team:

This means that they can make 5 complete teams of 8 children, with 3 children left over.

However, since one child goes home, there are only 42 children left. We can repeat the division:

This means that they can make 5 complete teams of 8 children, with 2 children left over. Therefore, they can make 5 complete teams of 8 children even after one child goes home.

The solution to the system of linear equations is (x, y) = (13, -4).

To solve the system of linear equations using Gaussian elimination, we need to eliminate one variable from one of the equations. Here, we can eliminate x from the second equation by subtracting twice the first equation from the second equation:

x + 2y = 5    (equation 1)

2x + 3y = 6    (equation 2)

--------------

  -2x - 4y = -10   (2 * equation 1)

        y = -4

Now, we can substitute the value of y into the first equation to solve for x:

x + 2(-4) = 5

x - 8 = 5

x = 13

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We can see here that an equation to show the total length of the bandages if they are placed end-to-end is:

([tex]1\frac{1}{4}[/tex] × [tex]2) + (1\frac{2}{4}[/tex] × 1) + ([tex]1\frac{3}{4}[/tex] × 3) + (2 × 4) + (3 × 6) = [tex]35\frac{1}{4}[/tex]

An equation in mathematics is a claim made regarding the equality of two expressions. Normally, it has two sides that are separated by an equal sign (=).

Variables, constants, and mathematical operations including addition, subtraction, multiplication, division, exponentiation, and more can be used on each side of the equation.

We can see here that the above answer is correct because on the number line:

[tex]1\frac{1}{4}[/tex] has 2 Xs on it.

[tex]1\frac{2}{4}[/tex] has 1 X on it.

[tex]1\frac{3}{4}[/tex] has 3 Xs on it.

2 has 4 Xs on it.

3 has 6 Xs on it.

And multiplying and adding the variables, we arrived at:  [tex]35\frac{1}{4}[/tex]

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Using Pythagorean theorem, none of the triangles given are right triangles.

To determine which of the triangles are right triangles, you can use the Pythagorean theorem (a² + b² = c²), where a and b are the shorter side lengths and c is the longest side (hypotenuse).

Triangle 1:
Side lengths: 12, 19, 505
Checking: 12² + 19² = 144 + 361 = 505 ≠ 505²
Triangle 1 is not a right triangle.

Triangle 2:
Side lengths: 16, 19, 1467
Checking: 16² + 19² = 256 + 361 = 617 ≠ 1467²
Triangle 2 is not a right triangle.

Triangle 3:
Side lengths: 14, 20, 596
Checking: 14² + 20² = 196 + 400 = 596 ≠ 596²
Triangle 3 is not a right triangle.

Triangle 4:
Side lengths: 11, 23, 1421
Checking: 11² + 23² = 121 + 529 = 650 ≠ 1421²
Triangle 4 is not a right triangle.

None of the triangles given are right triangles.

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The volume of the box is 160 cubic inches. Using the volume of a rectangular prism formula the volume of the box is 160 cubic inches. Using another formula base area times height, the area is the same.

The volume of James' rectangular prism box can be calculated using both unit cubes and a formula. Part (a) involves counting the unit cubes in the model and multiplying the number of cubes by their volume, which is 1 cubic inch. In this case, there are 160 unit cubes, so the volume of the box is 160 cubic inches.

Part (b) involves using the formula for volume of a rectangular prism, which is length times width times height. Plugging in the given dimensions, we get 8 x 4 x 5 = 160 cubic inches, which is the same as the answer in part (a) using unit cubes.

Part (c) involves using a different formula for volume, which is base area times height. In this case, the base of the rectangular prism is a rectangle with length 8 inches and width 4 inches, so the base area is 8 x 4 = 32 square inches. Multiplying by the height of 5 inches, we get 160 cubic inches, which is again the same as the answers in parts (a) and (b).

Using the volume formulas is much quicker and more efficient than counting unit cubes, especially for larger boxes. However, counting unit cubes can provide a more concrete visual representation of the volume and can be helpful for students who are just learning about volume. In this case, the answers obtained using the formulas were the same as the answer obtained by counting unit cubes, which reinforces the accuracy of the formulas.

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Answer:

3 / (7-5)

Step-by-step explanation:

"Subtract 5 from 7"

When you're subtracting from something, the reverse the order of the numbers.  So, the expression here would be "7 - 5"

"Then, divide 3 by the result."

Here, you're dividing 3 by the result, so the 3 must be in the numerator (on top of the fraction), and the "result from the previous step must be in the denominator (on the bottom of the fraction).  So, the expression here would be "3 / result"

Since order of operations forces division to happen before subtraction, we'll need parentheses around the first result to force the subtraction to happen first, as instructed.

So, the final expression would be "3 / (7-5)"

The volume of the 4th doll is approximately [tex]130.1 cm^3.[/tex]

The diameter of the smallest doll is 1.9 cm, so its radius is 0.95 cm (half of the diameter).

Similarly, the radius of the second smallest doll is (2.85/2) = 1.425 cm.

Since the scale factor is the same from one doll to the next, the ratio of the radius of the second smallest doll to the radius of the smallest doll is:

1.425 cm / 0.95 cm = 1.5

Similarly, the ratio of the radius of the third smallest doll to the radius of the second smallest doll is also 1.5.

Using this pattern, we can find the radius of the 4th doll as:

Radius of 4th doll = 1.5 × (Radius of 3rd doll) = 1.5 × 2.1375 cm = 3.2063 cm (rounded to 4 decimal places)

The volume of the 4th doll can then be calculated as:

Volume of 4th doll = (4/3) × π ×[tex](Radius of 4th doll)^3[/tex]

                           = (4/3) × π × [tex](3.2063 cm)^3[/tex]

                           ≈ [tex]130.1 cm^3[/tex]

Therefore, the volume of the 4th doll is approximately [tex]130.1 cm^3.[/tex]

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Answer:

Step-by-step explanation:

Since the two cylinders are similar, their corresponding dimensions (radius and height) are proportional. Let the radius of the smaller cylinder be r.

Then, we can write:

r / 6 = R / 30

where R is the radius of the larger cylinder.

Simplifying this equation, we get:

R = 5r

Now, we can use the formula for the volume of a cylinder to find the volume of the larger cylinder:

Volume of smaller cylinder = πr^2h = 90 cm^3

Volume of larger cylinder = πR^2H = π(5r)^2(30) = 750πr^2 cm^3

Substituting R = 5r, we get:

Volume of larger cylinder = 750πr^2 cm^3

Therefore, the volume of the larger cylinder is 750π times the volume of the smaller cylinder:

Volume of larger cylinder = 750π(90 cm^3) = 67,500π/ cm^3 (approx. 211,239.74 cm^3 rounded to five decimal places).

Answer:

Yes

Step-by-step explanation:

9/25

√9/√25 = 3/5 = 0.6

so it is a rational number because it has an integer as a denominator also because the decimal is not reoccurring

Answer: Yes

Step-by-step explanation:

Yes.

9/25 = .36  

Because the decimal stop it is rational

Only decimals that have no pattern and go on infinitely then it is irrational like [tex]\pi[/tex]  or √7   if you plug those into a calculator they go on forever and have no pattern

The correct answer is: b

8 points within any 36-month period

6 points within any 24-month period

4 points within any 12-month period

In most US states, drivers are assigned points for certain traffic violations or accidents. If a driver accumulates too many points within a certain period of time, they may be labeled as a "negligent operator" and face penalties such as license suspension or revocation. The point thresholds for being labeled as a negligent operator may vary by state, but the options given in the question are generally accurate.

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The maximum error of estimate is 4.75%.

To calculate the maximum error of estimate for the given problem, we will use the formula for margin of error:

Margin of Error = Z-score * (Standard Deviation / √n)

Where:
- Z-score corresponds to the 90% confidence level, which is 1.645
- Standard Deviation is 0.5 hours
- n is the sample size, which is 300 students

Margin of Error = 1.645 * (0.5 / √300) ≈ 0.0475

To express this as a percentage, multiply by 100:

0.0475 * 100 ≈ 4.75%

Thus, the maximum error of estimate with a 90% confidence level is 4.75%.

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Answer: 3/8

Step-by-step explanation:

7/8-1/2

Find common denominator which is 8 and you get that from 1/2 by multiplying 4 on the top and bottom and so you wind up with 4/8 and 7/8-4/8 is 3/8

The vector field for  7 (x,y) = xî + xyî (1,3), (-1,2), (3,-2) can be plotted with arrows with magnitude 4 at point (1,3), magnitude 3 at point (-1,2), and magnitude 3 at point (3,-2).

The  given vector field 7 (x,y) = xî + xyî (1,3), (-1,2), (3,-2) can be drawn by plotting arrows at each point (x,y) in the plane with the direction and magnitude of each arrow presented  by the vector

7 (x,y) = xî + xyî.

over point (1,3),

the vector is 7(1,3) = 1î + 3î = 4î.

over point (-1,2),

the vector is 7(-1,2) = -1î - 2î = -3î.

Over point (3,-2),

the vector is 7(3,-2) = 3î - 6î = -3î.

The vector field for  7 (x,y) = xî + xyî (1,3), (-1,2), (3,-2) can be plotted with arrows with magnitude 4 at point (1,3), magnitude 3 at point (-1,2), and magnitude 3 at point (3,-2).

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By applying the Consecutive Angles Theorem, Jim can confirm that the angled section for the steps is indeed a parallelogram, even without accurately proving that any two sides are parallel.

Jim building a deck with an angled section in the shape of a parallelogram. To be assured that he has an actual parallelogram, Jim can use the Consecutive Angles Theorem.

This theorem states that if the consecutive angles of a quadrilateral are supplementary (add up to 180 degrees), then the quadrilateral is a parallelogram.

To use the Consecutive Angles Theorem, Jim should follow these steps:

1. Measure the four angles of the quadrilateral he has created for the angled section of the deck.
2. Check if the consecutive angles are supplementary (i.e., the sum of each pair of consecutive angles is equal to 180 degrees).
3. If all consecutive angles are supplementary, he can be assured that the quadrilateral is a parallelogram.

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The general solution to the differential equation is, |3x^4/(y(x+y))|x + |x^3| ln|y| + |x^3| ln|x+y| + h(x) = C.

The partial derivative of (3xy + y^2) with respect to y is 6xy + 2y, and the partial derivative of (x^2 + xy) with respect to x is 2x + y. Since these are not equal, the differential equation is not exact.

To make it exact, we need to find an integrating factor μ(x, y) such that μ(x, y)(3xy + y^2) dx + μ(x, y)(x^2 + xy) dy = 0 is exact. We can find μ(x, y) by using the formula:

μ(x, y) = e^(∫(∂M/∂y - ∂N/∂x)/N dx)

where M = 3xy + y^2 and N = x^2 + xy. We have:

(∂M/∂y - ∂N/∂x)/N = (6xy + 2y - 2x - y)/(x^2 + xy) = (6xy - x - y)/(x^2 + xy)

We can now find the integrating factor μ(x, y) by integrating this expression with respect to x:

μ(x, y) = e^(∫(6xy - x - y)/(x^2 + xy) dx) = e^(3ln|x| - ln|y| - ln|x+y| + C) = e^(ln|x^3/(y(x+y))| + C) = |x^3/(y(x+y))|e^C

where C is the constant of integration.

Now we multiply the original differential equation by the integrating factor μ(x, y) to obtain:

|3x^4/(y(x+y))| dx + |x^3/(y(x+y))| dy = 0

This is now an exact differential equation, and we can find its solution by integrating with respect to x or y. Integrating with respect to x, we get:

|3x^4/(y(x+y))|x + g(y) = C

where g(y) is the constant of integration. To find g(y), we integrate the coefficient of dy:

g(y) = ∫|x^3/(y(x+y))| dy = |x^3| ln|y| + |x^3| ln|x+y| + h(x)

where h(x) is another constant of integration. Substituting g(y) back into the solution, we have:

|3x^4/(y(x+y))|x + |x^3| ln|y| + |x^3| ln|x+y| + h(x) = C

This is the general solution to the differential equation.

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The perimeter is indeed 26 feet, which means they have used all the extra fencing.

Let's start by assuming that the length and width of the garden patch are whole numbers of feet, since we are asked to use only whole numbers.

Let's call the length of the garden patch "L" and the width "W".

We know that Liam has 26 feet of fencing left over. This fencing will be used to make the perimeter of the garden patch, which is given by:

Perimeter = 2L + 2W

We can substitute the value of the perimeter with the amount of fencing that Liam has:

26 = 2L + 2W

Simplifying this equation, we get:

13 = L + W

Since we want to use all the extra fencing, we know that the perimeter of the garden patch must be 26 feet. We can use this information to write another equation:

Perimeter = 2L + 2W = 26

We can substitute the value of 13 for L + W in this equation:

2L + 2W = 26

2L + 2(13-L) = 26

2L + 26 - 2L = 26

26 = 26

This equation is true, which means that our assumption that L and W are whole numbers is correct.

Therefore, the dimensions of the garden patch that Liam and his brother can make for their sister are 6 feet by 7 feet.

To check, we can calculate the perimeter:

Perimeter = 2L + 2W = 2(6) + 2(7) = 12 + 14 = 26

So the perimeter is indeed 26 feet, which means they have used all the extra fencing.

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