20. The volume of a large crate is 84 yd³. It is 23 yd wide and 4 yd high. What is the length of the crate?​

The solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.

Equation is a mathematical statement that expresses the equality of two expressions. It shows the relationship between two or more variables and can be written using symbols, numbers, and operations. Equations are used to describe physical laws, to make calculations, and to solve problems. Examples of equations include the Pythagorean theorem, Newton's laws of motion, and linear equations.

We solve this differential equation using Laplace inverse, with the initial conditions y(0)=1 and y'(0)=-2. First, we take the Laplace transform of the equation:

L[y''+y]=L[t]

Using the properties of Laplace transform, we can write this as:

s2Y(s)-sy(0)-y'(0)+Y(s)= (1/s)

Substituting the initial conditions and rearranging terms, we have:

Y(s)= (1/s) + (2/s2) + (1/s2)

We can then invert the Laplace transform to get the solution of the original equation:

y(t)= 1-2t+te-t

Therefore, the solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.

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The equation of the normal plane is 4y = 4π, or equivalently, y = π.

The word osculate comes from the Latin osculatus, which is a past participle of the verb osculari, which means "to kiss." Thus, an osculating plane is one that "kisses" a submanifold.

To find the osculating plane and normal plane of the curve x = sin 2t, y = t, z = cos 2t at the point (0, π, 1), we need to follow these steps:

Step 1: Find the first and second derivatives of the curve with respect to t.

x' = 2cos2t

y' = 1

z' = -2sin2t

x'' = -4sin2t

y'' = 0

z'' = -4cos2t

Step 2: Evaluate the derivatives at t = π.

x'(π) = 2cos2π = 2

y'(π) = 1

z'(π) = -2sin2π = 0

x''(π) = -4sin2π = 0

y''(π) = 0

z''(π) = -4cos2π = -4

So the velocity vector at the point (0, π, 1) is v = ⟨2, 1, 0⟩, the acceleration vector is a = ⟨0, 0, -4⟩, and the curvature vector is κv = ⟨0, 4, 0⟩.

Step 3: Use the velocity and acceleration vectors to find the normal vector of the osculating plane.

The normal vector of the osculating plane is given by the cross product of the velocity and acceleration vectors:

n = v × a = ⟨2, 1, 0⟩ × ⟨0, 0, -4⟩ = ⟨4, 0, 0⟩

Step 4: Use the normal vector and the point (0, π, 1) to find the equation of the osculating plane.

The equation of the osculating plane is given by:

4(x - 0) + 0(y - π) + 0(z - 1) = 0

Simplifying, we get:

4x - 4 = 0

So the equation of the osculating plane is 4x = 4, or equivalently, x = 1.

Step 5: Use the curvature vector to find the normal vector of the normal plane.

The normal vector of the normal plane is given by the curvature vector:

n' = κv = ⟨0, 4, 0⟩

Step 6: Use the normal vector and the point (0, π, 1) to find the equation of the normal plane.

The equation of the normal plane is given by:

0(x - 0) + 4(y - π) + 0(z - 1) = 0

Simplifying, we get:

4y - 4π = 0

So, the equation of the normal plane is 4y = 4π, or equivalently, y = π.

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None of the given numbers make the equation 8/y² + 2 true.

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

To solve this problem, we can substitute each of the given numbers (0, 1, 2, 3, 4) for y in the equation 8/y² + 2 and see if the equation is true.

Substituting y=0 would make the denominator of the fraction zero, which is undefined, so y=0 is not a valid choice.

Substituting y=1 would give us:

8/1² + 2 = 8 + 2 = 10

So, 1 is not the answer.

Substituting y=2 would give us:

8/2² + 2 = 8/4 + 2 = 2 + 2 = 4

So, 2 is not the answer.

Substituting y=3 would give us:

8/3² + 2 = 8/9 + 2 = 0.888 + 2 = 2.888

So, 3 is not the answer.

Substituting y=4 would give us:

8/4² + 2 = 8/16 + 2 = 0.5 + 2 = 2.5

So, 4 is not the answer.

Therefore, none of the given numbers make the equation 8/y² + 2 true.

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In triangle ABC

a) The value of x = 29⁰

b) The angle c equal to 93⁰

A triangle is a closed plane figure that is formed by connecting three line segments, also known as sides, at their endpoints. The three endpoints, or vertices, where the sides of the triangle meet are not collinear. Triangles are important in mathematics and geometry because they are the simplest polygon that can exist in two-dimensional space.

In a triangle, the sum of all interior angles is always 180 degrees. Therefore, we can use this fact to find the value of x and angle c.

We know that:

angle a = 35⁰

angle b = 52⁰

angle c = 3(x+2)⁰

Using the fact that the sum of all interior angles in a triangle is 180 degrees, we can write:

angle a + angle b + angle c = 180

Substituting the values we know, we get:

35 + 52 + 3(x+2) = 180

Simplifying the equation, we get:

87 + 3x + 6 = 180

3x + 93 = 180

3x = 87

x = 29

Therefore, x = 29⁰

To find angle c, we can substitute the value of x into the equation we were given for angle c:

angle c = 3(x+2)

angle c = 3(29+2)

angle c = 3(31)

angle c = 93

Therefore, angle c is equal to 93⁰.

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Answer:

1629.24m

Step-by-step explanation:

starting with the easy ones

1) Rectangle 1:

Surface area of rectangle=Length x width

SAR= 24x21

= 504m

2) Rectangle 2:

SAR= 19x21

=399

3) Rectangle 3

SAR= 21x8

=168

4) Rectangle 4:

SAR= 21x11

=231

*because the top and bottom are trapeziums the formular for it is

A=1/2(a+b)h

although those trapeziums don't have h(Height)

it needs to be broken down into two triangles and a rectangle. to find the height*

5) side/height of triangle A:

formula: C squared= a squared + b squared

in this case we already have C and A. meaning we have to rearrange the formula to:

x = c^2 - a^2

x = 8^2 - 2.5^2

x = sqrt 57.75

x = 7.61

6) Trapezium

SA= 1/2(a+b)h

SA= 1/2(19+24)7.61

=163.62

7) add all surface area together

which should equal 1629.24m

Answer:

[tex]3( {2}^{2} ) - {2}^{2} + 4 = 12[/tex]

[tex] {2}^{3} + b( {2}^{2} ) + 43(2) - 126 = 4b - 204[/tex]

[tex]4b - 32 = 12[/tex]

[tex]4b = 44[/tex]

[tex]b = 11[/tex]

For this value of b, these graphs will intersect at (2, 12). Please use your graphing calculator to confirm that this is the only point of intersection.

Answer:

5.4

Step-by-step explanation:

square of 29 = 5.385164807

5.385164807 estimated is 5.4

Answer:

finding the perimeter, you sumthe distance all round that is 7+7.5+17.8+6=38.3

38.3 is the perimeter

Suppose that the function h is defined as follows. -2 -1 h(x)= if – 2, the graph is given below: (see image)

A graph is a mathematical structure used to represent relationships between objects or entities. It consists of a set of vertices (also known as nodes) and a set of edges that connect pairs of vertices.

In a graph, the vertices represent the objects or entities being studied, while the edges represent the connections or relationships between them.

For example, in a social network graph, the vertices might represent individual users, and the edges might represent their connections (e.g. friendships) with other users.

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One per 2,500 can be used to represent the rate of 80cm to 2km.

To write 80cm to 2km as a rate, we need to convert the units to the same system. We can convert 80cm to kilometers by dividing by 100,000 (since there are 100,000 centimeters in a kilometer):

80 cm/100,000 = 0.0008 km

Now we can express the rate as:

0.0008 km per 2 km

Or we can simplify it by dividing both the numerator and denominator by 0.0008:

1 per 2,500

Therefore, 80cm to 2km can be expressed as a rate of 1 per 2,500.

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If a researcher is standing in a corner (point c) of a triangle plot. the angle to point a is S76E .  the distance between point A and point B is 210 ft.

To find the distance between points A and B, we can use the Law of Cosines. Let's first label the angles of the triangle:

   Angle ACB = 180 - (76 + 32) = 72 degrees

   Angle BAC = 76 degrees

   Angle ABC = 32 degrees

Using the Law of Cosines, we have:

AB^2 = AC^2 + BC^2 - 2(AC)(BC)cos(ACB)

where;

AB is the distance between points A and B

AC is the distance between points A and C (210 ft

BC is the distance between points B and C (150 ft)

ACB is the angle between AC and BC (72 degrees)

Plugging in the values, we get:

AB^2 = 210^2 + 150^2 - 2(210)(150)cos(72)

AB^2 = 44100

AB = sqrt(44100)

AB = 210 ft

Therefore, the distance between point A and point B is 210 ft.

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The composite functions (f/g)(x) and (f-g)(x) are (2x-1)/√(3x-5) and (2x-1) -√(3x-5)

To calculate (f/g)(x), we need to divide f(x) by g(x):

(f/g)(x) = f(x)/g(x) = (2x-1)/√(3x-5)

The domain of (f/g)(x) is the set of all x-values for which the denominator √(3x-5) is not equal to zero and non-negative

3x-5 ≥ 0, or x ≥ 5/3

Therefore, the domain of (f/g)(x) is x ≥ 5/3.

To calculate (f-g)(x), we need to subtract g(x) from f(x):

(f-g)(x) = f(x) - g(x) = (2x-1) - √(3x-5)

The domain of (f-g)(x) is the set of all x-values for which the expression inside the square root is non-negative:

3x-5 ≥ 0, or x ≥ 5/3

Therefore, the domain of (f-g)(x) is x ≥ 5/3.

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the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center would be 3 times the original length of AB.

How to solve the question?

To find the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center, we can use the following formula:

AB' = AB x 3

where AB' is the length of AB after the dilation, and AB is the original length of AB.

However, we need to first determine the length of AB in the original right triangle ABC. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Let's assume that AB is the hypotenuse of the right triangle ABC, and that AC and BC are the other two sides. Then we have:

AB²= AC²+ BC²

Without more information about the lengths of AC and BC, we cannot determine the value of AB. However, once we have determined the length of AB, we can use the formula above to find the new length of AB after dilation.

Assuming we know the length of AB in the original right triangle ABC, we can now use the formula for dilation to find the new length of AB:

AB' = AB x 3

For example, if AB is 5 units long in the original triangle, then after dilation, AB' would be:

AB' = 5 x 3 = 15 units

Therefore, the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center would be 3 times the original length of AB.

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Answer: t=2.59

Step-by-step explanation:

This is a matter of clearing out the equation

set 1151=800e^0.14t

1151/800=e^0.14t

ln(1151/800)/0.14=t

t=2.59

Answer:

3,4,5 is the answer

Step-by-step explanation:

for the explanation using pythagoras theoem

[tex] {3}^{2} + {4}^{2} = {5 \\ }^{2} \\ 3 \times 3 + 4 \times 4 = 5 \times 5 \\ 9 + 16 = 25 \\ 25 = 25[/tex]

may you give me branliest as you promised

Arun's purple paint will be redder.

Percentage is a way of expressing a proportion or a fraction as a number out of 100. It is represented by the symbol "%". For example, if you say that 20% of students in a class scored an A grade in a test, it means that 20 out of every 100 students received an A grade.

Deondra mixed 6 cups of blue paint and 7 cups of red paint, so the percent of red paint in her mixture is:

7 / (6 + 7) × 100% = 53.8%, which rounds to 54%.

Arun mixed 2 cups of blue paint and 3 cups of red paint, so the percent of red paint in his mixture is:

3 / (2 + 3) × 100% = 60%.

Since Arun's mixture has a higher percentage of red paint, his purple paint will be redder than Deondra's.

Therefore, the answer is Arun's purple paint will be redder.

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Hence, to close the expansionary gap, the government would need to decrease spending by $100 billion.

To close the expansionary gap, the government would need to decrease spending by $100 billion.

The formula to calculate the change in equilibrium output due to a change in government spending is:

∆Y = (∆G / (1 - MPC))

Where:

∆Y = change in equilibrium output

∆G = change in government spending

MPC = marginal propensity to consume

Here, the output is $400 billion above its natural level, and the central bank agrees to adjust the money supply to hold the interest rate constant, so there is no crowding out. Therefore, we can assume that the change in government spending (∆G) would have a one-to-one effect on the equilibrium output (∆Y).

Given that MPC = 3/4, we can plug in the values into the formula:

400 = (∆G / (1 - 3/4))

∆G = (1 - 3/4) * 400

∆G = (1/4) * 400

∆G = 100

Hence, to close the expansionary gap, the government would need to decrease spending by $100 billion.

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Answer:

B

Step-by-step explanation:

THE TWO SMALLER SIDES = THE LARGER SIDE HENCE ITS A TRIANGLE

Answer: 4.8%

Step-by-step explanation: the total amount of pens in the drawer is (10+12+3)  = 25

the amount of red pens in the drawer is 3

the probability of picking out a red pen from the drawer = 3/25

the amount of blue pens in the drawer is 10

the probability of picking out a red pen from the drawer = 10/25

the probability of picking out a red pen then a blue pen afterwards = (10/25 x 3/25) = 4.8%

The correct option -D. 5;  Thus, the value of x for the given external secant segment and the tangent on the circle is found as:x = 5.

A line that precisely intersects a circle at two points is said to be a secant.

The size of the angle created when two tangents, two secants, or two tangents cross outside of a circle is equal to one-half a positive difference between the sizes of the intercepted arcs.

Using the Theorem:

The square of a length of tangent is equal the the product of such external secant segment and the overall length of the secant if one secant and one tangent both drawn to a circle from a single exterior point:

4(4 + x) = 6²

16 + 4x = 36

4x = 36 - 16

4x = 20

x = 20/4

x = 5

Thus, the value of x for the given external secant segment and the tangent on the circle is found as:x = 5.

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Answer:

  (d)  25 feet and 20 feet​

Step-by-step explanation:

You want the possible dimensions of a pool with an area of 500 square feet and a perimeter of 90 feet.

The area is the product of the length and width. The area of the pools offered in the answer choices are ...

  (a)  15·30 = 450 . . . square feet

  (b)  10·35 = 350 . . . square feet

  (c)  50·10 = 500 . . . square feet

  (d)  25·20 = 500 . . . square feet

The area requirement eliminates answer choices A and B.

The perimeter is twice the sum of length and width. The perimeters of the possible pools are ...

  (c) 2(50 +10) = 120 . . . feet

  (d) 2(25 +20) = 90 . . . feet

The perimeter requirement eliminates answer choice C.

The pool's possible length and width are 25 feet and 20 feet, choice D.

__

Additional comment

You could write a quadratic equation for the pool dimensions, but doing that will generally involve more work than checking the given answer choices.

If x is the width, then 45-x is the length, and the area is ...

  x(45 -x) = 500

  x² -45x +506.25 = -500 +506.25 . . . multiply by -1, complete the square

  x = 22.5 -√6.25 = 20 . . . . take the square root; width is the smaller dimension

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The appropriate measurement unit for force is  kg.m/s²

A force is simply referred to as either a push or pull of an object resulting from the object's interaction with another object.

From Newton's Second Law, force is expressed as;

F = m × a

Where is mass of object and a is the acceleration

One Newton is defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared.

Since F = m × a

Plug in m = kg and a = m/s²

F = kg × m/s²

F = kg.m/s²

Therefore, the unit of measurement is kg.m/s².

Option B is the correct answer.

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The range of the relation whose graph is shown include the following: C. -1 ≤ y ≤ 1.

In Mathematics and Geometry, a domain is the set of all real numbers for which a particular function is defined.

Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = {-4, 4} or -4 ≤ x ≤ 4.

Range = {-1, 1} or -1 ≤ y ≤ 1.

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Using the total area of the rectangular pool we know that the required width of the pathway is 3.6 ft respectively.

In the Euclidean plane, a rectangle is a quadrilateral with four right angles.

A parallelogram with a right angle or an equiangular quadrilateral, where equiangular means that all of its angles are equal, are other ways to define it.

A rectangle with four equal-length sides is a square.

Squares are not always rectangles, but rectangles are always squares.

So, the pool and walkway together have a 360 square foot total space.

Assume that the walkway is w feet in width.

Since the pool has an additional w feet of width on both sides, the total area's measurements can be expressed as (8 + 2w) by (10 + 2w).

Thus, we can construct the following equation:

(8 + 2w) x (10 + 2w) = 360

By enlarging and condensing the left side, we obtain:

4w² + 36w - 80 = 0

Using the quadratic formula, we can find w:

w = (-b ± √(b² - 4ac)) / 2a

Here, an equals 4, b equals 36, and c equals -80. When these values are added to the formula, we obtain:

w = (-36 ± √(36² - 4(4)(-80))) / 8

w = (-36 ± √(1872)) / 8

w ≈ -5.6 or w ≈ 3.6

We can ignore the first option because the width cannot be negative. Consequently, the pathway should be around 3.6 feet wide.

Therefore, using the total area of the rectangular pool we know that the required width of the pathway is 3.6 ft respectively.

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The coordinates of the vertices of the image rectangle M'P'A'T' are:

M'(2,1), P'(3,1), A'(3,3), T'(2,3).

What is rectangle?

A rectangle is a geometric shape that has four sides and four right angles (90 degrees) with opposite sides being parallel and equal in length.

To find the coordinates of the vertices of the image rectangle M'P'A'T', we need to apply a transformation to each vertex of the original rectangle MPAT.

We can see that the original rectangle MPAT has sides parallel to the x and y-axes, which suggests that it is aligned with the coordinate axes. We can also see that the length of its sides are equal, which means it is a square.

To transform this square, we can use a combination of translations, rotations, and reflections. However, since we don't have any information about the type of transformation that is being applied, we can assume that the simplest transformation is a reflection across the line y=x.

To reflect a point (x,y) across the line y=x, we swap its x and y coordinates to get the reflected point (y,x). Therefore, the coordinates of the vertices of the image rectangle M'P'A'T' are:

M'(2,1)

P'(3,1)

A'(3,3)

T'(2,3)

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There are 52 quarters and 15 half-dollars

Let's represent the number of half-dollars as "x" and the number of quarters as "y".

From the problem statement, we know that:

x + y = 67 (because there are a total of 67 coins)

y = 3x + 7 (because the number of quarters is 7 more than three times the number of half-dollars)

We can use substitution to solve for x:

x + (3x + 7) = 67

4x + 7 = 67

4x = 60

x = 15

So there are 15 half-dollars. We can use this to find the number of quarters:

y = 3x + 7

y = 3(15) + 7

y = 52

So there are 52 quarters.

Therefore, there are 52 quarters and 15 half-dollars.

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Answer:

1/8

Step-by-step explanation:

If "over" refers to (4 * 6) as a whole,

[tex]\frac{3}{4*6}\\\\=\frac{3}{24}\\\\=\frac{1}{8}[/tex]

If "over" refers to division,

[tex]3 \div 4 \times 6\\= 0.75 \times 6\\= 4.5[/tex]

I would suppose it's the first one, so, ignore the second one if you haven't learnt BODMAS yet.

Hope this helps and be sure to mark this as brainliest! :)

The negation is the fourth option.

6 + 3 ≠ 9 or 6 - 3 ≠ 9

The negation of an equation is an inequality such that we just change the equal sign, by the "≠" sign.

Here we start with the two equations.

6 + 3 = 9 or 6 - 3 = 9

Just change the equal signs for different signs:

6 + 3 ≠ 9 or 6 - 3 ≠ 9

That is the negation, fourth option.

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The circumference and approximate area of the given​ circle is 69.08 inches & 380.14 square inches.

What is circumference?

Circumference is the distance around the edge of a circular object or a round shape. It is the length of the boundary or perimeter of the circle. The formula is given by C = 2πr, where C is the circumference, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14.

The circumference of a circle is given by the formula:

C = 2πr

where r is the radius of the circle and π (pi) is a mathematical constant approximately equal to 3.14.

Using this formula and plugging in the given value of radius:

C = 2 x 3.14 x 11

C = 69.08 inches (rounded to two decimal places)

So the circumference of the circle with 11 inches radius is approximately 69.08 inches.

The area of a circle is given by the formula:

A = πr²

Again, using the given value of radius and approximating π to 3.14:

A = 3.14 x 11²

A = 3.14 x 121

A = 380.14 square inches (rounded to two decimal places)

So the approximate area of the circle with 11 inches radius is approximately 380.14 square inches.

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Answer:

Axis of symmetry =1

vertex =(1,2)

y intercept =0

min/max= -6,2

domain= 0,1,2

range =y≥1,2