Find an equation for the line below. ​

The volume of the toy is 35cm^3. The solution has been obtained using the formula of triangular pyramid.

What is a triangular pyramid?

A triangular pyramid has a triangle-shaped base, and its apex is where the three triangular faces converge. There is a unique type of triangular pyramid called a tetrahedron, which features equilateral triangles on each face. The formula for a triangular pyramid consists of the volume and surface area of the pyramid, which are used to determine the height and slant height as well as the three triangular-shaped sides.

In the given figure, x represents the base(b), y represents the base height(h) and z represents the height(H).

Let's first find the pyramid's base area(B) which is given by;

⇒B = 1/2 * b * h

⇒B = 1/2 * 3 * 7

⇒B = 10.5cm^2

The volume of the toy is given by:

⇒V = 1/3 * B * H

⇒V = 1/3 * 10.5 * 10

⇒V = 35cm^3

Hence, the volume of the toy is 35cm^3.

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The angular speed of 14 revolutions per second is equal to 87.96 radians per second.

Angular speed is a measure of how fast an object is rotating or revolving around an axis. It is typically measured in units of angular displacement per unit of time, such as degrees per second, revolutions per minute (RPM), or radians per second.

To convert angular speed from revolutions per second (RPS) to radians per second (RAD/s), we can use the following formula:

Angular speed (RAD/s) = 2π x Revolutions per second (RPS)

Given that the angular speed is 14 revolutions per second, we can substitute this value into the formula:

Angular speed (RAD/s) = 2π x 14 RPS

Angular speed (RAD/s) = 87.96 RAD/s

So, 14 revolutions per second is equivalent to 87.96 radians per second.

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What is ratio?

A ratio is a way of comparing two or more values or quantities. It expresses the relationship between the size of one value or quantity to the size of another. Ratios are often written as fractions or as ":".

The Federal Reserve is making nickels with a constant of proportionality. This can be determined by analyzing the relationship between the amount of nickels made and the time elapsed. If the number of nickels made is directly proportional to the time elapsed, then the ratio between the number of nickels made and the time elapsed will be constant.

As we can see from the table, the ratio between the number of nickels made and the time elapsed is constant, which is 6000 nickels per hour. This means that the Federal Reserve is making nickels with a constant of proportionality.

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The Federal Reserve is making nickels with a constant of proportionality.

What is ratio?

A ratio is a way of comparing two or more values or quantities. It expresses the relationship between the size of one value or quantity to the size of another. Ratios are often written as fractions or as ":".

The Federal Reserve is making nickels with a constant of proportionality. This can be determined by analyzing the relationship between the amount of nickels made and the time elapsed. If the number of nickels made is directly proportional to the time elapsed, then the ratio between the number of nickels made and the time elapsed will be constant.

As we can see from the table, the ratio between the number of nickels made and the time elapsed is constant, which is 6000 nickels per hour. This means that the Federal Reserve is making nickels with a constant of proportionality.

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Answer:

a)  No inverse function

[tex]\textsf{b)} \quad c^{-1}(x)=5+\sqrt{x-2},\;\; c\geq2[/tex]

[tex]\textsf{c)} \quad f^{-1}(x)=-\dfrac{2x}{3x-5}[/tex]

Step-by-step explanation:

Given function:

[tex]a(x)=-4x^2+3[/tex]

To find the inverse of the function, swap x and y:

[tex]\implies x=-4y^2+3[/tex]

Rearrange the equation to isolate y:

[tex]\implies x-3=-4y^2[/tex]

[tex]\implies y^2=-\dfrac{x-3}{4}[/tex]

[tex]\implies y^2=\dfrac{3-x}{4}[/tex]

As the domain and range of function a(x) are unrestricted, the inverse relation is a “sideways” parabola and therefore it is not a function, since it fails the vertical line test.

Given function:

[tex]c(x)=(x-5)^2+2,\;\;x\geq5[/tex]

As the domain of function c(x) is restricted to x ≥ 5, the range is also restricted.

To find the inverse of the function, swap x and y:

[tex]\implies x=(y-5)^2+2[/tex]

Rearrange the equation to isolate y:

[tex]\implies x-2=(y-5)^2[/tex]

[tex]\implies \pm\sqrt{x-2}=y-5[/tex]

[tex]\implies y=5\pm\sqrt{x-2}[/tex]

As the domain of function c(x) is restricted to x ≥ 5 then:

[tex]\implies y=5+\sqrt{x-2}[/tex]

Replace y with c⁻¹(x):

[tex]\implies c^{-1}(x)=5+\sqrt{x-2}[/tex]

The domain of the inverse of a function is the same as the range of the original function.  Therefore, the domain of the inverse function is restricted to x ≥ 2.

Given function:

[tex]f(x)=\dfrac{5x}{3x+2}[/tex]

To find the inverse of the function, swap x and y:

[tex]x=\dfrac{5y}{3y+2}[/tex]

Rearrange the equation to isolate y:

[tex]\implies x(3y+2)=5y[/tex]

[tex]\implies 3xy+2x=5y[/tex]

[tex]\implies 3xy-5y=-2x[/tex]

[tex]\implies y(3x-5)=-2x[/tex]

[tex]\implies y=-\dfrac{2x}{3x-5}[/tex]

Replace y with f⁻¹(x):

[tex]\implies f^{-1}(x)=-\dfrac{2x}{3x-5}[/tex]

The required word problem of expression  2(5+w)=42, can be given as twice the sum of a number, and 5 is 42,

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

Here,
The question seems to be incomplete, from sources complete expression is given as 2(5+w)=42,

So word problem of expression 2(5+w)=42 can be given as,
Twice the sum of a number and 5 is 42, when we sum a number with 5 and doubled that number gives 42 as a product.

Thus, the required word problem of expression  2(5+w)=42, can be given as twice the sum of a number, and 5 is 42,

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The expression to determine the distance between the lowest and the highest point is -

{x} = 14505 + |- 282|

Given is that the highest elevation in California is 14,505 feet, and the lowest elevation is − 282 feet.

The expression to determine the distance between the lowest and the highest point can be written as -

{x} = 14505 + |- 282|

where -

|  |  represents modulus function.

Therefore, the expression to determine the distance between the lowest and the highest point is -

{x} = 14505 + |- 282|

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We can use the integral  x=4secθ  for each of the indefinite integrals below.

In the given question we can use either 4tanθ,4sinθ or 4secθ to substitute and simplify the given indefinite integrals,

the 1st integral is,

∫ √(x²-16) dx

we can use the trigonometric substitution x=4secθ in this integral, due to which after simplification it will become as,

∫16 secθtan²θ dθ

which then can easily be solved.

the 2nd integral is,

∫ x²/√(x²-16) dx

we can use the trigonometric substitution x=4sinθ in this integral, due to which after simplification it will become as,

∫16 sin²θ dθ

which then can be solved by using a reduction formula.

the 3rd integral is,

∫ dx /(16+x²)³

we can use the trigonometric substitution x=4tanθ in this integral, due to which after simplification it will become as,

∫(1/16⁴) cos⁴θ dθ

the 4th integral is,

∫ (x²-16)^(5/2)dx

we can use the trigonometric substitution x=4secθ in this integral, due to which after simplification it will become as,

∫4096 secθtan⁶θ dθ

the 5th integral is,

∫  dx /(16-x²)^(3/2)

we can use the trigonometric substitution x=4sinθ in this integral, due to which after simplification it will become as,

∫(1/16² cos²θ) dθ

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this is ur answer.

written by me

I am also class 10 student

Solve for the first variable in one of the equations, then substitute the result into the other equation.Point Form:(2,−1)(2,-1)Equation Form:x=2,y=−1

so the sanswer is 19

                              —
                               3.

The solutions that match the given differential equation are a)y=0 and b)y=2x.

The differential equation is a homogeneous linear differential equation with constant coefficients, which can be written in the form of y" + p(x)y' + q(x)y = 0. The general solution to this type of equation is y = c1e^(rx) + c2e^(rx) where r is the root of the characteristic equation r^2 + p(x)r + q(x) = 0.

In this case, the equation is of the form xy'' - y' = 0. By dividing both sides by x, we get y'' - (1/x)y' = 0, which is a homogeneous linear differential equation with constant coefficients. The characteristic equation is r^2 - (1/x)r = 0. The roots of this equation are r1 = 0 and r2 = 1/x.

Therefore, the general solution to this differential equation is y = c1 + c2x.

y=0 is a solution of the differential equation since it satisfies the equation when plugged in.

y=2x is also a solution of the differential equation since it also satisfies the equation when plugged in.

y=2x^2 and y=2 are not solutions of the differential equation because when plugged into the equation they don't satisfy it.

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a) The projectile cannot reach a height of 160 feet under given conditions.

b) The projectile takes a time of 2 second to hit the ground.

In this question we find the case of a project under free fall, that is, a vertical uniformly accelerated motion due to action of gravity, whose formula is shown below:

s = - 16 · t² + v₀ · t

Where:

Part A - If we know that v₀ = 32 ft / s and s = 160 ft, then the time associated is:

- 16 · t² + v₀ · t - s = 0

- 16 · t² + 32 · t - 160 = 0

By quadratic formula:

t₁ = 1 + i 3, t₂ = 1 - i 3

It is physically impossible to reach a height of 160 feet under conditions given.

Part B - If we know that s = 0 ft and v = 32 ft / s, then the final time is:

- 16 · t² + v₀ · t - s = 0

- 16 · t² + 32 · t = 0

- 16 · t · (t - 2) = 0

The final time for the projectile is 2 seconds.

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As per the unitary method, there are 10 chairs and there are 2 tables.

In math the term called unitary method is known as  a process of finding the value of a single unit, and based on this value we can find the value of the required number of units.

Here we have given that tom and jerry have one day to work, but two tasks to focus on building chairs and tables.

Here we also know that  if tom spends all day building chairs, then he will make 16 chairs.

And if he instead devotes his day to building tables, then tom will make 4 tables.

Where as if jerry spends his day building chairs, then he will make 14 chairs on the other hand if he spends the day building tables, then he will make 7 tables.

Based on the given details, here by using the unitary method, for one day, he will product 10 chairs and 2 tables.

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Mrs. Barton should use 2 cups of peanut butter in her huge batch with 1 cup of oil.

What is fraction?

Fractions are a way of representing part of a whole. They are written as two numbers separated by a slash. The top number is called the numerator, and the bottom number is called the denominator. For example, in the fraction 1/2, 1 is the numerator and 2 is the denominator. This fraction represents one part of a total of two parts.

Mrs. Carter's famous peanut butter cookies call for 1 cup of peanut butter for every 1/2 of a cup of oil. Today, she wants to make a huge batch with 1 cup of oil.

From the above question:

1/2 cup of oil = 1 cup of peanut butter

1 cup of oil = x

Cross Multiply

1/2x = 1 × 1

x = 1 ÷ 1/2

x = 1 × 2/1

x = 2 cups of peanut butter.

Therefore, for huge batch with 1 cup of oil, she should use 2 cups of peanut butter.

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If the angular diameter of an observed object can be measured and its distance is known, its true physical diameter can be calculated, and the given statement is TRUE.

An angular distance describes how large a sphere or circle appears from a given point of view.

The angular diameter is calculated as:

Angular Diameter=206265X(Actual diameter/Distance)

The angular diameter of an object is the angle the object makes as seen by an observer. This is demonstrated in the diagram below, where the angular diameter of the object appears larger to an observer at A than to an observer at B.

Angular diameter can also refer to the distances between two objects, measured on the celestial sphere.

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The absolute change in average television sizes from the time of Joseph's deployment time he returned is given as follows:

2.4 inches.

The relative change is given as follows:

5.94%.

The absolute change is given by the absolute value of the difference of the final value by the initial value, hence:

|42.8 - 40.4| = 2.4 inches.

The relative change is given by the division of the absolute change by the initial measure, and then multiplied by 100%, hence:

2.4/40.4 x 100% = 5.94%.

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Check the picture below.

Answer:

9

Step-by-step explanation:

(-2+5)^2

3^2

9

The arrangement with descending trend is 1.247,1.244,1.234

A decimal number is a number that consists of a whole number and a fractional part separated by a point. For example – 3.5, 6.79, 78.32, etc.

Given here: The decimal numbers 1.234,1.244,1.247

Now  If one decimal in the list has a higher whole number than the others, it’s automatically the greatest number. If a decimal in the list has a smaller whole number than the others on the list, it’s automatically the smallest number. . Create enough rows and columns for all of the digits and number spaces (including the decimal point). Going from left to right, label the columns as tens, ones, decimal, tenths, hundredths, and thousandths.

Thus,

 ones  decimals tenths Hundreths thousands

   1           .              2           3                 4

   1           .              2           4                 4

   1           .              2           4                  7

∴ The order from greatest to least is 1.247,1.244,1.234

Hence, The arrangement with the descending trend is 1.247,1.244,1.234

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The order of the functions' average rates is h(x), g(x) and f(x)

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

The functions

The interval is given as

4 ≤ x ≤ 7

And the average rate is calculated as

Rate = (f(x₂) - f(x₁))/(x₂ - x₁)

So, we have

f(x) rate = (10 + 10)/(7 - 4) = 6.67

g(x) rate = (19 - 7)/(7 - 4) = 4

For h(x), we have

h(7) = 7^2 - 9 * 7 + 5 = -9

h(4) = 4^2 - 9 * 4 + 5 = -15

So, we have

h(x) rate = (-15 + 9)/(7 - 4) = -2

When compared, we have

-2 less than 4 less than 6.67

So, the order is h(x), g(x) and f(x)

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Answer:

1. e its given

2. d

3. b

4. c

5. a

this all are proved by the theorem

Answer:

x^3+1/x^3 = x^2

Step-by-step explanation:

If x^2+1/x^2 =1

The equation x^2+1/x^2 =1 states that the sum of the square of x and the reciprocal of x squared is equal to 1. We can multiply both sides of this equation by x(x+1/x) which is x^2 + 1.

This gives us (x^2)(x+1/x) = x^3 + 1/x^3 = 1.

So, x^3+1/x^3 = x^2.

that x^2+1/x^2 =1, we can deduce that x^3+1/x^3 = x^2. This is because we multiplied both sides of the first equation by x^2 + 1/x^2,

which is equal to x^3+1/x^3 = x^2

From the given option , the completed sentence is : An odd function is symmetric with respect to the origin .

The Odd Functions are functions which return the negative inverse when x is replaced with "-x" . which means that function f(x) is an odd function when f(-x) = -f(x) .

The Odd Functions are symmetrical about origin because ; the Odd function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.

Therefore , the odd functions are symmetric about the origin .

The given question is incomplete , the complete question is

Select the correct choice that completes the sentence below.

An odd function is symmetric with respect to the _____  .

(a) y-axis

(b) origin

(c) line y = x

(d ) x-axis.

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Answer:

I believe it's 350 R(remainder) 7

The amount of water placed in the fountain is given as follows:

D. 9,600π in³

The volume of a cylinder of radius r and height h is given by the equation presented as follows:

V = πr²h.

The parameters for the fountain in this problem are given as follows:

r = 30 in, h = 16 in.

Then the volume of the entire fountain is given as follows:

V = π x 30² x 16

V = 14,400π in³.

Two thirds of this entire volume is given as follows:

2/3 x  14,400π in³ = 9,600π in³

Meaning that the correct option is given by option D.

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Answer:

see below

Step-by-step explanation:

every stage is 60m

stage one = -340+60 = -280

stage 2: -280 +60 = -220

stage 3: -220 +60 = -160

stage 4: -160+60 = -100

stage 5: -100+60= -40

stage 6: -40+60 =20

it'll take it 6 stages to fully surface

Answer:

x =~ 21.93

Step-by-step explanation:

x = hypotenusa

x² = (18 / 2)² + 20²

x² = 9² + 400

x² = 81 + 400

x² =~ 481

x = [tex]\sqrt{481}[/tex]

x =~ 21.93

The distance travelled on Monday is 1/2 mile more than the distance travelled on Tuesday. Yes, the answer is reasonable, the difference between the two numbers indicates the remaining distance.

In mathematics, a fraction is used to denote a portion or component of the total. It stands for the proportionate pieces of the whole. Numerator and denominator are the two components that make up a fraction.

Given that the distance travelled on Monday is 5/6 miles, and the distance travelled on Tuesday is 1/3 miles.

To find how much more distance is covered on Monday:

x= 5/6 - 1/3  

x= 5/6 - 2/6

x= 3/6

x= 1/2

Hence, the distance travelled on Monday is 1/2 mile more than the distance travelled on Tuesday.

Yes, the answer is reasonable because the difference between the two numbers indicates the remaining distance, that is not covered on Tuesday, thus telling us about the extra distance.

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