find the second linearly independent soln. of the DE from the first
x^2y'' - 42y = 0; y1=x^7

Step-by-step explanation:

There are   pi radians in 180 degrees

  you want   1/4 of 180 degrees   (45/180 = 1/4)  

    so this would be   1/4 pi radians  or   pi / 4   radians

45 *   pi / 180 = pi/4  

The system of inequalities for the graph is

a) y < 3x - 4

b) y = x + 3

Given data ,

Let the inequality equations be represented as A and B

where

Let the first point be P ( 0 , 3 )

Let the second point be Q ( -3 , 0 )

Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )

m = ( 3 - 0 ) / ( 0 + 3 )

m = 1

Now , equation of line is y - y₁ = m ( x - x₁ )

y - 0 = 1 ( x + 3 )

y = x + 3

And , the second equation is

Let the first point be R ( 0 , -4 )

Let the second point be S ( 2 , 2 )

Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )

m = ( -4 - 2 ) / ( 0 - 2 )

m = -6/-2

m = 3

Now , equation of line is y - y₁ = m ( x - x₁ )

y - 2 = 3 ( x - 2 )

Adding 2 on both sides , we get

y = 3x - 4

Since , it is a dotted line ,

y < 3x - 4

And , the solution to the inequality is point of intersection

where A ( 3.5 , 6.5 ) is the solution

Hence , the inequality equations are solved and A ( 3.5 , 6.5 ) is the solution

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Answer:

20

Step-by-step explanation:

She biked an equal amount each day for 8 days to a total of 32 miles. We can write that as 8x = 32. 32/8 = 4 so x = 4. To find how much shell bike in 5 days, we multiply it by x(4). 5*4 = 20.

1/4π radians is approximately equal to 45 degrees.

To convert a value from radians to degrees, multiply the value by 180/π. For example, to convert π/4 radians to degrees, multiply π/4 by 180/π to get 45 degrees. This conversion is useful for converting angles between the two units of measurement.

Degrees = Radians × 180/π.

Therefore, to convert 1/4π radians to degrees, we have:

Degrees = (1/4π) × (180/π) ≈ 45°

Hence, 1/4π radians is approximately equal to 45 degrees.

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The calculated number of plants the flower bed can contain is 120

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

Dimensions = 16 m by 9 m

So, the area of the flower bed is

Area = 16 * 9

Evaluate

Area =  144

Also, we have

Each plant requires a space of 1.2m x 1m?

This means that

Plant area = 1.2 * 1

Plant area = 1.2

So, we have

Plants = 144/1.2

Evaluate

Plants = 120

Hence, the number of plants is 120

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Answer:

Step-by-step explanation:

To calculate the number of plants that can be planted in the rectangular flower bed, we need to calculate the area of the flower bed and divide it by the space required for each plant.

The area of the flower bed is calculated by multiplying its length and breadth.

So, the area of the flower bed is 16m x 9m = 144 sq.m.

Each plant requires a space of 1.2m x 1m = 1.2 sq.m.

Therefore, the number of plants that can be planted in the flower bed is:

144 sq.m. ÷ 1.2 sq.m./plant = 120 plants.

So, you can plant 120 plants in the rectangular flower bed.

The coordinates in the solution to the systems of inequalities is (12, 1)

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

y > -4x + 6

y > 1/3x - 7

Next, we plot the graph of the system of the inequalities

See attachment for the graph

From the graph, we have solution to the system to be the shaded region

The coordinates in the solution to the systems of inequalities graphically is (12, 1)

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Therefore, Jan has taken the wrong slant height, he has taken height, h is a place of slant height, l. That's why he has got the wrong answer.

Given:,

r= d/2

22/2= 11m

Using the Pythagorean theorem, we can find the length:

l =[tex]11\sqrt{5}\\ = 24.6m[/tex]

Therefore, to find the surface area:

SA= 3.14(270.56 + 121)

=[tex]1229.5m^2[/tex]

Therefore, to find the volume:

V= [tex]1/3 \pi.r^2h[/tex]

V= [tex]2786.23m^3[/tex]

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The Complete Question

Jan INCORRECTLY finds the surface area of the cone using the following work. Explain Jan's error and find the correct volume AND surface area of the cone. 22 m SA = = url + ar?

The amount of poster board left over will be 1275 square inches, the equation used is  A = (b₁ + b₂)h/2.

To calculate the amount of poster board that will be left over after Ole cuts out a piece of poster board that is the same size as the window,

we need to find the area of the trapezoid window and subtract it from the area of the poster board.

The formula to find the area of a trapezoid is A = (b₁ + b₂)h/2, where b₁ and b₂ are the lengths of the parallel sides and h is the height.

Let's assume that the length of the top parallel side of the trapezoid is 20 inches, the length of the bottom parallel side is 10 inches, and the height is 15 inches.

Using the formula, we get A = (20 + 10) x 15 / 2 = 225 square inches. This is the area of the window.

To find the area of the poster board, we multiply the length and width, which gives 25 x 60 = 1500 square inches.

Finally, we subtract the area of the window from the area of the poster board using the equation:

1500 - 225 = 1275 square inches.

Therefore, the amount of poster board left over will be 1275 square inches.

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The complete question is

Ole has a window that has the dimensions and shape like the trapezoid shown below. He also has a large rectangular piece of poster board that measures 25 inches by 60 inches. If Ole cuts out a piece of poster board that is exactly the same size as the window, which equation can be used to calculate the amount of poster board that will be left over?

Step-by-step explanation:

40 :100        yellow to purple,  divide both sides by 20

2:5

Answer:

the ratio of yellow to purple is 40:100 that is 2:5 in the simplest form.

Step-by-step explanation:

Hope it helps.

If the height of the cylinder is 10, the diameter of the base of the cylinder is 8 units.

A right circular cylinder has two congruent circular bases and a curved surface area. To find the diameter of the base, we need to first find the radius of the base.

Let's begin by using the given information to write an equation. The surface area of a cylinder is given by the formula:

Surface Area = 2πr² + 2πrh

where r is the radius of the base and h is the height of the cylinder.

We are given that the surface area of the cylinder is 112π and the height is 10. Substituting these values, we get:

112π = 2πr² + 2πr(10)

Simplifying the equation by dividing both sides by 2π, we get:

56 = r² + 10r

Now we have a quadratic equation in terms of r. We can solve this equation by using factoring, completing the square, or the quadratic formula. Let's use factoring:

r² + 10r - 56 = 0

(r + 14)(r - 4) = 0

The solutions are r = -14 and r = 4. Since the radius cannot be negative, the radius of the base is 4.

The diameter of the base is twice the radius, so the diameter is 8.

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Answer:

7, 14

Step-by-step explanation:

Look at the first column of numbers: 2.8 m in 2 min.

We can get a unit rate from those numbers:

2.8 m / 2 min = 1.4 m/min

He walks at a rate of 1.4 m/min

Now multiply the unit rate by each number of minutes to find the number of meter.

Let's do the second column as a check: time = 3 min

1.4 m/min × 3 min = 4.2 m

4.2 m is the number on the table, so our unti rate is correct.

5 min:

1/4 m/min × 5 min = 7 m

10 min:

1.4 m/min × 10 min = 14 m

The correct step to prove that [tex]BC^2 = AB^2 + AC^2[/tex] is:

By the cross product property, [tex]AC^2 = BC \cdot AD[/tex].

To prove that [tex]BC^2 = AB^2 + AC^2[/tex], we can use the triangle similarity and the Pythagorean theorem. Here's a step-by-step explanation:

Given triangle ABC with right angle at A and segment AD perpendicular to segment BC.

By triangle similarity, triangle ABD is similar to triangle ABC. This is because angle A is common, and angle BDA is a right angle (as AD is perpendicular to BC).

Using the proportionality of similar triangles, we can write the following ratio:

[tex]$\frac{AB}{BC} = \frac{AD}{AB}$[/tex]

Cross-multiplying, we get:

[tex]$AB^2 = BC \cdot AD$[/tex]

Similarly, using triangle similarity, triangle ACD is also similar to triangle ABC. This gives us:

[tex]$\frac{AC}{BC} = \frac{AD}{AC}$[/tex]

Cross-multiplying, we have:

[tex]$AC^2 = BC \cdot AD$[/tex]

Now, we can substitute the derived expressions into the original equation:

[tex]$BC^2 = AB^2 + AC^2$\\$BC^2 = (BC \cdot AD) + (BC \cdot AD)$\\$BC^2 = 2 \cdot BC \cdot AD$[/tex]

It was made possible by cross-product property.

Therefore, the correct step to prove that [tex]BC^2 = AB^2 + AC^2[/tex] is:

By the cross product property, [tex]AC^2 = BC \cdot AD[/tex].

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The samples are {(H, R), (H, W), (H, B), (T, R), (T, W), (T, B)}. Thus, the correct option is D.

Given that:

A coin is tossed.

A spinner has 3 equal sections: Red, White, and Blue.

The samples are groups of clearly specified components. The number of items in a finite set is denoted by a curly bracket.

The total number of samples is calculated as,

Total samples = 2 x 3

Total samples = 6

The samples are {(H, R), (H, W), (H, B), (T, R), (T, W), (T, B)}.

Thus, the correct option is D.

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The correct statement regarding the measure of variability used to represent the data is given as follows:

The IQR of 10 is the most accurate to use, as the data is skewed.

The interquartile range of a data-set is given by the difference of the third quartile by the first quartile of the data-set.

It is the measure of variability to be used when a data-set contains outliers, or is skewed, as is the case for this problem.

The charity received 18 donations, hence the quartiles are given as follows:

Hence the IQR is given as follows:

IQR = 22.5 - 12.5

IQR = 10.

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The correct matrix representation is;  [tex]\left[\begin{array}{ccc}1&5\\3&2\end{array}\right][/tex]

Based on the coordinates of a vector, We can represent a vector pointing at a point by its x coordinate, and y coordinate,

Consider that there are two points represented by their x-, y coordinates as P₁(x₁,y₁) P₂(x₂,y₂)

Given here the points are P₁(1, 3) and P₂(5, 2)

Thus, by the coordinates of the two vectors, we can represent the matrix  ;

[tex]\left[\begin{array}{ccc}1&5\\3&2\end{array}\right][/tex]

Hence, Option A) is the correct answer.

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Answer:

d = [tex]\sqrt{58}[/tex]

Step-by-step explanation:

calculate the distance d using the distance formula

d = [tex]\sqrt{x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = (2, 5 ) and (x₂, y₂ ) = (- 5, 8 )

d = [tex]\sqrt{(-5-2)^2+(8-5)^2}[/tex]

   = [tex]\sqrt{(-7)^2+3^2}[/tex]

   = [tex]\sqrt{49+9}[/tex]

   = [tex]\sqrt{58}[/tex]

Answer:

1) B = 88°, a = 13.1, b = 26.9

2) C = 73°, a = 20.9, b = 12.6

Step-by-step explanation:

To solve for the remaining sides and angles of the triangle, given two sides and an adjacent angle, use the Law of Sines formula:

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]

Given values:

As the interior angles of a triangle sum to 180°:

[tex]\implies A+B+C=180^{\circ}[/tex]

[tex]\implies B=180^{\circ}-A-C[/tex]

[tex]\implies B=180^{\circ}-29^{\circ}-63^{\circ}[/tex]

[tex]\implies B=88^{\circ}[/tex]

Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:

[tex]\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

[tex]\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]

Solve for a:

[tex]\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]

[tex]\implies a=\dfrac{24\sin 29^{\circ}}{\sin 63^{\circ}}[/tex]

[tex]\implies a=13.0876493...[/tex]

[tex]\implies a=13.1[/tex]

Solve for b:

[tex]\implies \dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]

[tex]\implies b=\dfrac{24\sin 88^{\circ}}{\sin 63^{\circ}}[/tex]

[tex]\implies b=26.9194211...[/tex]

[tex]\implies b=26.9[/tex]

[tex]\hrulefill[/tex]

Given values:

As the interior angles of a triangle sum to 180°:

[tex]\implies A+B+C=180^{\circ}[/tex]

[tex]\implies C=180^{\circ}-A-B[/tex]

[tex]\implies C=180^{\circ}-72^{\circ}-35^{\circ}[/tex]

[tex]\implies C=73^{\circ}[/tex]

Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:

[tex]\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

[tex]\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]

Solve for a:

[tex]\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]

[tex]\implies a=\dfrac{21\sin 72^{\circ}}{\sin 73^{\circ}}[/tex]

[tex]\implies a=20.8847511...[/tex]

[tex]\implies a=20.9[/tex]

Solve for b:

[tex]\implies \dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]

[tex]\implies b=\dfrac{21\sin 35^{\circ}}{\sin 73^{\circ}}[/tex]

[tex]\implies b=12.5954671...[/tex]

[tex]\implies b=12.6[/tex]

Answer:

Eat that cookie and you will not regret about it

The difference in temperature between Rome and Munich is given as follows:

6ºC.

The difference in temperature between Rome and Munich is given by the subtraction of Rome's temperature by Munich's temperature.

The bar graph in the context of this problem gives the temperature for each town.

From the bar graph given by the image presented at the end of the answer, the temperatures for Rome and Munich are given as follows:

Hence the difference in temperature between Rome and Munich is given as follows:

11 - 5 = 6ºC.

The graph is given by the image presented at the end of the answer.

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Answer:

The Value of Y is -8

Step-by-step explanation:

I believe it is -8 because giving the function, y = 8 - 2x, given that x = 8, we can replace the value into a function. y = 8 - 2(8) = 8 - 16 = -8.

So therefore, the value of y is -8. Hopes this helps :p

The volume of the cylindrical shaped pool with water is V = 75.4 feet³

Given data ,

Let the diameter of the pool be d = 8 feet

So , radius r = 4 feet

And , the height of the cylindrical pool is h = 3 feet

Now , the water is half full

So , Volume of Cylindrical pool is V = ( 1/2 ) πr²h

V = ( 1/2 ) ( 3.14 ) ( 4 )² ( 3 )

On simplifying , we get

V = 75.4 feet³

Hence , the amount of water in pool is V = 75.4 feet³

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Approximately 5048 sports cars will have less than 150,000 miles on the odometer

Mean = 120,000

Standard deviation = 30,000

Total group = 6000

Total Miles = 150,0000

An odometer is a device used to gauge how far a wheeled vehicle has driven. The tool might be mechanical, electrical, or a hybrid of the two.

Calculating the Z score -

z = x - u/a

= 15,0000 - 12,0000/30000

= 30000/30000

= 1

Using the standard normal table, to determine the value -

P ( Z<1) = 0.8413

Calculating the total number of cars -

= P value x total group of cars

= 6000 x 0.8413

= 5047.7 or 5048

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Answer:

60 ft^2

Step-by-step explanation:

To get the total volume we will need to multiply all the lengths. This means we will have to do 4 x 3x 5.

4 x 3 x 5 = 15x 4 = 60

Answer:

60 ft³

Step-by-step explanation:

V = 4 ft × 3 ft × 5 ft

V = 12 ft² × 5 ft

V = 60 ft³

Answer:

147

Step-by-step explanation:

V = h * S = h * Pi R^2 = h * Pi * d^2/4 = 7.5 * 3.14 * 25 / 4 = 147.188

Answer:

Step-by-step explanation:

148 people

The outside border of the garden would have walked around approximately  58%.

Without more information about the shape and dimensions of the garden, it's impossible to give an exact answer. However, if we assume that the garden is a rectangle, we can use the formula for the perimeter of a rectangle to estimate the percentage of the garden's border that would be walked around.

Let's say that the length of the garden is L and the width of the garden is W. The perimeter of the rectangle is then:

P = 2L + 2W

If the person walks from point A to point B along the outside of the garden, they are essentially walking along two sides of the rectangle. Let's call these sides S1 and S2. Depending on the location of A and B, S1 and S2 may be two adjacent sides, two opposite sides, or one side and one diagonal.

To estimate the percentage of the garden's border that the person would walk around, we can calculate the length of S1 and S2 and divide by the total perimeter of the rectangle:

Percentage walked = (S1 + S2) / P * 100%

Again, without more information about the shape and dimensions of the garden, we can't give an exact answer. However, if we assume that the person walks along two adjacent sides of the rectangle, the percentage of the garden's border that they would walk around would be:

Percentage walked = (2L + W) / (2L + 2W) * 100%

Simplifying this expression, we get:

Percentage walked = (2L + W) / (2(L + W)) * 100%

Assuming that L and W are measured in the same units (e.g. meters), we can simplify further:

Percentage walked = (2 + W/L) / (2 + 2W/L) * 100%

For example, if the length of the garden is 10 meters and the width of the garden is 5 meters, then the percentage of the garden's border that the person would walk around if they walked along two adjacent sides would be:

Percentage walked = (2 + 5/10) / (2 + 2*5/10) * 100%

= 7/12 * 100%

= 58.3%

So the outside border of the garden would have walked around approximately  58%.

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