How do you write the ratio 12:3 in simplest form

The volume of the triangular prism is 168 cm³ while the surface area of the rectangular and triangular prism are 286 cm² and 270 ft² respectively

An equation is an expression that shows the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on

The surface area of a solid figure is gotten by calculating each area of the surface and then summing.

a) For the rectangular prism:

Surface area = 2(9 * 7) + 2(9 * 5) + 2(7 * 5) = 286 cm²

b) For the triangular prism:

Surface area = 2(0.5 * 12 * 5) + (7 * 12) + (7 * 13) + (5 * 7) = 270 ft²

3) Volume  = (1/3) * base area * height = (1/3) *  (8 * 9) * 7 = 168 cm³

The volume of the triangular prism is 168 cm³

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Answer:

Neither

Step-by-step explanation:

Parallel slopes are the same and perpendicular slopes are opposite reciprocals which would be like 2/3 and -3/2.

The area at right end point is Rn=7.484 and area at left end points is Ln=7.476

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here f(x) = [tex]\frac{1}{x^3}[/tex] over the interval [1,5] and n=10 then

Δx = [tex]\frac{b -a}{n} = \frac{5-1}{10} = \frac{4}{10} = \frac{2}{5}[/tex]

Δx = 0.4

Then subinterval are, [0,0.4],[0.4,0.8],[0.8,1.2],[1.2,1.6],[1.6,2],[2,2.4],[2.4,2.8],[2.8,3.2],[3.2,3.6],[3.6,4]

Then, area using right end point is

Rn = Δx[f(0.4)+f(0.8)+f(1.2)+f(1.6)+f(2)+f(2.4)+f(2.8)+f(3.2)+f(3.6)+f(4)]

=> Rn = 0.4[[tex]\frac{1}{0.4^3} +\frac{1}{0.8^3} +\frac{1}{1.2^3} +\frac{1}{1.6^3} +\frac{1}{2^3} +\frac{1}{2.4^3} +\frac{1}{2.8^3}+\frac{1}{3.2^3} +\frac{1}{3.6^3}+\frac{1}{4^3}[/tex]]

=> Rn=0.4[18.71]

=> Rn= 7.484

Now area using  Left end point is

=>Ln = Δx[f(0)+f(0.4)+f(0.8)+f(1.2)+f(1.6)+f(2)+f(2.4)+f(2.8)+f(3.2)+f(3.6)]

=> Ln = 0.4[[tex]\frac{1}{0^3}+\frac{1}{0.4^3} +\frac{1}{0.8^3} +\frac{1}{1.2^3} +\frac{1}{1.6^3} +\frac{1}{2^3} +\frac{1}{2.4^3} +\frac{1}{2.8^3}+\frac{1}{3.2^3} +\frac{1}{3.6^3}[/tex]

=> Ln = 0.4[18.69]

=> Ln =7.476

Hence the area at right end point is Rn=7.484 and area at left end points is Ln=7.476

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Answer: 7, 7, 19, and 19

Recall that a rectangle has congruent parallel sides, meaning the sides opposite are equivalent in length. Therefore, 2 sides are both 12 cm longer than the other two sides. Let's write an equation:

Let x be the shorter side, and x + 12 be the longer side.

x + x + (x + 12) + (x + 12) = 52

Combine like terms.

4x + 24 = 52

4x = 28

x = 7

Therefore, the shorter sides are 7, and the longer sides are 19

Answer:

4800 liters fuel to travel 600 kilometers

Answer:

50L

Step-by-step explanation:

Given ,

To Find -

Now,

If Car travels 240 km in 20 L of petrol

Then,

The car travels 1 km in 20/240 L of petrol.

Then,

Petrol needed to travel 600 km

20/240 × 600

20/24 × 60

20/4 × 10

5 × 10

= 50 L

Hence, 50 L of petrol is needed to travel 600 km.

let f be a differentiable function such that f(1)=2 and f′(x)=x2 2cosx 3 then f(4)= 4.1414

if a function is differential at x if the limit of a function exists at a pint x and the function is continuous at x point also then the function is differentiable at the point x

since sin and cos function are differentiable and their limit exists

Given that f is  a function given by

f(x) =2cosx +1

Here x is taken in radians.

Hence when x =1.5

we have cos 1.5 = 0.0707

2cosx = 0.1414

And hence

2cosx+1=1+0.1414

=1.1414

Thus we get

f(1.5) = 1.1414 apprxy

f( 4) = 0.1414 +4

      = 4.1414

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Answer:

C) 10.790

Step-by-step explanation:

we are given:

[tex]f(1)=2\\[/tex]

[tex]\frac{dy}{dx} =\sqrt{x^2+2cos(x)+3}[/tex]

we need to find:

[tex]f(4)=?[/tex]

integrate the function by using a TI-84 calculator (math > 9:fnInt( > input)

[tex]\\\int\limits^4_1 {\sqrt{x^2+2cos(x)+3} } \, dx \\\\\\=8.789[/tex]

since we are looking on the interval [1, 4] we need to add our two limiting values together:

[tex]F(4)+F(1)=2+8.789\\\\=10.790[/tex]

The correct statements are,

⇒ Vertical reflection and translation.

Reflection is a type of transformation that flip a shape in mirror line, so that each point is the same distance from the mirror line as its reflected point.

We have to given that;

Triangles STU and S'T'U' are congruent.

Now, We know that;

Reflection is a type of transformation that flip a shape in mirror line, so that each point is the same distance from the mirror line as its reflected point.

And,  The translated shapes (or the image) appear to be the same size as the original shape, indicating that they are congruent. They've simply shifted in one or more directions.

Hence, The correct statement to Triangles STU and S'T'U' are congruent are,

⇒ Vertical reflection and translation.

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Positive rational numbers cannot be added together.

If real numbers are multiplied, they are closed.

Irrational numbers cannot be divided in any way.

Subtraction causes closed integers.

Positive rational numbers cannot be added together.

a) The set is closed since the product of any two real numbers is a real number.

b) The quotient could occasionally be logical. (18/2) Equals 3, for instance. The set is thus not closed.

c) The set is closed since the difference between any two numbers is also an integer.

d) The set is closed since the product of any two positive rational numbers is a positive rational number.

The complete question is

Select from the drop-down menus to correctly identify whether the given operation is closed or not closed with respect to each set of numbers.

Real numbers are Closed/Not Closed under multiplication.

Irrational numbers are Closed/Not Closed under division.

Integers are Closed/Not Closed under subtraction.

Positive rational numbers are Closed/Not Closed under addition.

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This is a second order linear ordinary differential equation (ODE) of the form Xy″ (3x – 1)y′ – (4x 9)y is 0.

To solve this equation, we can use the method of integrating factors. The goal is to convert the given equation into a first-order linear ODE which can be easily solved using the technique of separation of variables.

The first step is to find the integrating factor, which is a function that when multiplied with the given equation, makes it a first-order linear ODE. In this case, the integrating factor is given by

[tex]= > e^{\int(3x-1) dx} = e^{(3x^2/2 - x)}[/tex]

Next, we multiply both sides of the given equation with the integrating factor to get

[tex]= > e^{(3x^2/2 - x)} = xy" (3x - 1)y' - (4x - 9)y = 0.[/tex]

After integrating both sides with respect to x, we get an equation in the form of a first-order linear ODE.

Finally, we solve the first-order linear ODE to obtain the general solution for y, which can then be solved using the initial condition y(0) = 0 to get the particular solution for y.

The method of integrating factors is a useful technique for solving second-order linear ODEs.

Complete Question:

Solve the differential equation xy"+(3x−1)y′−(4x+9)y=0 by using Laplace Transform.

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From the Histogram shown below , the number of values in the data set that are greater than 5.5 but less than 8.5 are 17 .

A Histogram is defined as a graphical representation of data that displays the frequency distribution of a set of continuous or discrete data. It shows how data is distributed across a range of values, by dividing the data into a set of bins and then plotting the number of data points that fall into each bin as bars.

From the Histogram we see that between 5.5 and 8.5 there are 3 bars, that means the three heights are 7, 5, and 5.

So , to find the number of values ,  we add these three heights:

that means ⇒ 7 + 5 + 5 ⇒ 17.

Therefore , there are 17 values in the data set that are greater than 5.5 but less than 8.5  .

The given question is incomplete , the complete question is

Given the following histogram for a set of data, how many values in the data set are greater than 5.5 but less than 8.5 ?

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Answer:

The area of the rectangle is  [tex]8\sqrt{6}+\sqrt{18}[/tex]  ft².

Step-by-step explanation:

The formula for the area of a rectangle is length times width

[tex]A=lw[/tex]

We are given

[tex]l=8+\sqrt{3}[/tex]

[tex]w=\sqrt{6}[/tex]

Given this information we can solve for the area.

[tex]A=(8+\sqrt{3})*\sqrt{6}[/tex]

Use the Distributive Property to distribute [tex]\sqrt{6}[/tex] over [tex]8+\sqrt{3}[/tex].

[tex]A=8\sqrt{6}+\sqrt{6} \sqrt{3}[/tex]

We can simplify this to

[tex]A=8\sqrt{6}+ 3\sqrt{2}[/tex]  or  [tex]A=8\sqrt{6}+\sqrt{18}[/tex]

The total meter reading of Dan electricity usage from July to October including the meter reading as at July is 33,289 units

Quantity of electricity Dan used from July to October = 942 units

Meter reading in July= 32, 347 units

The total meter reading = Quantity of electricity Dan used from July to October + Meter reading in July

= 942 units + 32, 347 units

= 33,289 units

Therefore, Dan used a total of 33,289 units of electricity from July to October including the previous reading.

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Answer:

Below

Step-by-step explanation:

( 2/15 is incorrect)

1/12 is the same as   1 ÷ 12    use a calculator to see = .8333........

 do the same with the others and you will se that 7/8 = .875  and is the only one that terminates

The fraction 7/8 is the one that results in a terminating decimal.

A decimal representation of a fraction results in a terminating decimal if and only if all the prime factors of the denominator are 2 and/or 5, since 2 and 5 are the prime factors of 10, the base of our number system.

Let's analyze each fraction's denominator:

- For the fraction 1/12, the denominator is 12. The prime factors of 12 are 2 and 3. So, this fraction does not result in a terminating decimal.

- For the fraction 4/9, the denominator is 9. The prime factor of 9 is 3. Hence, this fraction doesn't result in a terminating decimal.

- For the fraction 2/15, the denominator is 15. The prime factors of 15 are 3 and 5. As the denominator contains the prime number 3, it doesn't result in a terminating decimal.

- For the fraction 7/8, the denominator is 8. The only prime factor of 8 is 2, which is one of the prime factors of 10. This means that the decimal representation of this fraction will be a terminating decimal.

Therefore, the fraction 7/8 is the one that results in a terminating decimal.

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Kate need 3/4 cup of flour. She only ha two measuring cup, a 1/3 and a 1/8. The cup should Kate use is 1/8 .

The cup of flour she need is 3/4 that is 75% of the use.

There are two cups available 1/3 and 1/8

In percentages 1/3 can be 33.33%

And in percentages 1/8 can be 12.5%

The amount to be filled is 3/4 that is 75% so the cup that will be used for this is 1/8 that is 12.5% so that,

= 12.5 x 6

= 75

So we used 1/8 cup for 6 times we can get the 3/4 cup of flour.

A fraction represents a portion of a total. This entire may refer to a place or a group of places. The Latin word "fractio," which meaning "to break," is the source of the English term "fraction." The distribution of food and supplies as well as the lack of a metal currency were among the mathematical issues that the Egyptians utilized fractions to solve because they were the first civilization to understand fractions.

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Answer:

9

Step-by-step explanation:

7 x (9 x 2) = (7 x 9) x 2  This is the associative property.

7 x ( 18) = (63) x 2

126 = 126

The given coordinates make a right angled triangle.

Scaling is defined as the process of changing the dimensions of the original figure as per some specific proportional rule.

If the initial length is equivalent to {a}. Then, after the scaling by the scale factor of {K}, the length becomes {Ka}. Similarly, if the initial coordinates are (x, y) then after scaling the coordinates would be {Kx, Ky}.

Scale factor is a dimensionless quantity that tells by how much time a specific dimension of a figure is enlarged or reduced. Mathematically, it is the ratio of two similar quantities. In case of length scaling, we can write -

{K} = L{final}/L{initial}

Given are the coordinates of the triangle as -

P(-1, 3)

Q(9, - 1)

R(-3, - 2)

The distance formula is given as -

d² = (x₂ - x₁)² + (y₂ - y₁)²

PQ = 2√29

QR = √145

RP = √29

QR² = RP² + PQ²

145 = 29 x 4 + 29

145 = 145

Therefore, the given coordinates make a right angled triangle.

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The vector v with a magnitude of 9 and in the same direction as u = <18 / √29, 45 / √29>.

What is the vector?

A vector quantity has both a magnitude and a direction. Speed only has a magnitude, but no direction. Velocity has both.

Given a vector u = <2, 5>, the magnitude of the vector v that is in the same direction as u and has a magnitude of 9 can be found as follows:

Step 1: Find the unit vector in the direction of u

A unit vector in the direction of u can be found by dividing each component of u by its magnitude:

magnitude of u = √(2^2 + 5^2) = √(4 + 25) = √29

unit vector in the direction of u = u / magnitude of u = <2 / √29, 5 / √29> = <2 / √29, 5 / √29>

Step 2: Multiply the unit vector by the desired magnitude

To find the vector v with a magnitude of 9 and in the same direction as u, we can multiply the unit vector by 9:

v = magnitude * unit vector in the direction of u = 9 * <2 / √29, 5 / √29> = <18 / √29, 45 / √29>

Therefore, the vector v with a magnitude of 9 and in the same direction as u = <18 / √29, 45 / √29>.

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The population of a city can be modeled by a radical function, P(x) = 55,000 √x - 1945, where x represents the year and P(x) represents the population in that year.

The year 1945 has been subtracted from x in order to simplify the equation and make it easier to interpret.

In this particular problem, we are asked to find the year in which the population of the city was 275,000. To do this, we need to solve for x in the equation:

55,000 √x - 1945 = 275,000

The first step is to isolate x on one side of the equation. We can do this by adding 1945 to both sides:

55,000 √x = 275,000 + 1945

Next, we need to get rid of the square root symbol. One way to do this is to square both sides of the equation:

x = (275,000 + 1945) / 55,000^2

The square root has been eliminated, but the equation still doesn't give us x in a form that is easy to interpret. To get x in a more useful form, we can take the square root of both sides:

√x = √((275,000 + 1945) / 55,000^2)

So, the year in which the population of the city was 275,000 is approximately x + 1945 = 1976.

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Answer:

72 sq ft

Step-by-step explanation:

The area of the square pool is 6ft * 6ft = 36 sq.ft.

The perimeter of the pool is 2 * (6ft + 6ft) = 24ft.

The length of the gravel path is equal to the perimeter of the pool, so 24ft.

The width of the gravel path is 3ft, so the total area of the gravel path is 24ft * 3ft = 72 sq.ft.

Since the corners of the path are rounded, we can assume that the loss of area is negligible and assume that the area of the gravel path is 72 sq.ft.

So, the approximate area of the gravel path is 72 sq.ft.

The geometric form known as a pentagon has five sides and five angles.

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The only value of c that satisfies the initial condition g(2) = 5 is c = -3.

The graph of a function f consists of three line segments, with points (1,2), (2,3), and (3,4).

An antiderivative of f is a function g such that g'(x) = f(x). That is, g is the "opposite" of the derivative of f. The values of c in an antiderivative g(x) = f(x) + c are determined by the initial condition g(2) = 5, since g'(2) = f(2) = 3.

So, in order to determine the value of c, we need to integrate f(x) and find the value of g(2). The integral of f(x) is

g(x) = x² + 2x + c

Substituting x = 2 into this equation, we get

g(2) = 4 + 4 + c = 8 + c

Now, since g(2) = 5, we can solve for c:

5 = 8 + c

c = -3

Therefore, the only value of c that satisfies the initial condition g(2) = 5 is c = -3.

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The total number of student that plotted the number of book they read is 18.

A dot blot the caption Book Read Over the Summer goes from 0 to 7. 0 ha 1 dot, 1 ha 3 dot, 2 ha 6, 3 ha 2, 4 ha 3, 5 ha 1 dot, 6 ha 2, and 7 ha 0 dot are the numbers. Using a dot plot

the total number of students who tracked how many books they read

add total dots  

1 dot + 3 dot +  6 dot +  2 dot + 3 dot +1 dot + 2 dot + 0 dot = 18.

The total number of student that plotted the number of book they read is 18.

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As per the given joint density function, the marginal density function for y1 is f1(y1) = 1/y1, where y1 > 0.

Joint density functions are used to describe the distribution of two or more variables. They provide the probability of observing certain values for multiple variables at the same time.

For the given joint density function

[tex]= > f(y1, y2) = e^{-(y1 y2)}, y1 > 0, y2 > 0[/tex]

and 0 elsewhere, we are asked to find the marginal density function for y1, which is denoted by f1(y1).

Here the marginal density function of y1 is found by integrating the joint density function over all possible values of y2. It gives us the probability density for y1 alone, without considering the value of y2.

In order to find f1(y1), we integrate f(y1, y2) with respect to y2, from 0 to infinity:

=> f1(y1) = ∫ f(y1, y2)dy2

[tex]= > \int f(y1, y2) = \int {e^{-(y1 y2)},dy2}[/tex]

[tex]= -(1/y1)e^{-(y1 y2)}[/tex]

= −(1/y1) * 0 + (1/y1) = 1/y1

Here it is important to note that the marginal density function must always be non-negative, since it represents a probability density.

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The total amount of energy released or absorbed by the reaction, which is the change in free energy.

ΔG = ΔH - TΔS

This equation, also known as the Gibbs Free Energy equation, is used to calculate the change in free energy (ΔG) resulting from a chemical reaction. The equation states that the change in free energy is equal to the change in enthalpy (ΔH) minus the product of the absolute temperature (T) and the change in entropy (ΔS).

The enthalpy change (ΔH) is the change in the amount of energy released or absorbed during a reaction at constant pressure. It is a measure of the amount of energy stored in the bonds of the reactants and products. The entropy change (ΔS) is the measure of the randomness or disorder of a system. It is a measure of how much energy is dispersed or spread out from the reaction.

By combining the enthalpy and entropy changes of a reaction, the Gibbs Free Energy equation can be used to calculate the total amount of energy released or absorbed by the reaction, which is the change in free energy. For example, if ΔH is -50 kJ and ΔS is +40J/K, then the change in free energy is -10 kJ.

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Based on the graph, in year 4 there were 1.16 times more burglaries if compared to year 1.

Based on the graph, the number of burglaries these years was:

This shows a growing trend in the number of burglaries over the years.

number in year 4/ 1

255/220 = 1.159 which can be rounded to 1.16

Based on this, year 4 had 1.16 times more burglaries than year 1.

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The body will reach maximum height of 20 meter.

Law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time.

K + U = K' + U'

K = initial kinetic energy

U = initial potential energy

K' = final kinetic energy

U' = final potential energy

Given,

Mass of body m = 500g = 0.5 kg

Velocity v = 72km/h = 72000/3600 = 20 m/s

At initial point

Height h = 0

So, Potential energy U = mgh = mg(0) = 0

Kinetic energy K = (1/2)mv²

At highest point velocity u = 0

So, Kinetic energy K' = (1/2)u² = (1/2)×0 = 0

Potential energy U' = mgH

Now,

By law of conservation of energy

K + U = K' + U'

0 + (1/2)mv² = mgH + 0

(1/2)mv² = mgH

v² = 2gh

H = v²/2g

g is acceleration due to gravity, g = 10 m/s²

h = 20²/2×10

h = 20 meter

Hence, 20 meter is the maximum height to which the body will reach.

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The domain and range of x-intercept (600,0) is Domain: {600}, Range: {0}

The domain and range of y-intercept (0,300) is Domain: {0}, Range: {330}

What is the x-intercept?

The intersection of a function or equation's graph with the x axis is known as the x intercept. This can be compared to a point having a value of 0.

For any given curve, the x-intercept is a coordinate that is plotted on the x-axis. To put it another way, the x-intercept is the value of the x coordinate at the point where the graph crosses the x-axis, or we can say that it is the value of the x coordinate at a point where the value of the y coordinate equals zero.

Let  s= x, y=a

5x+10y=3000

x-intercept:

y=0

5x+10(0) = 3000

5x=3000

=> x = 3000/5 = 600

x-intercept is (600,0)

y-intercept:

x=0

5(0)+10y = 3000

10y=3000

=> y = 3000/10 = 300

y-intercept is (0,300)

Plot these points and join them

graph attached.

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The average velocity of the function y = x² + 8x over the interval [5, 8] is 21.

Average velocity is a measure of the rate of change of an object's position over time. It is calculated by dividing the total change in position (displacement) of an object by the time interval over which the change occurred.

In mathematical terms, average velocity is given by the formula: v_avg = Δx / Δt, where Δx is the change in position (displacement) of the object and Δt is the time interval over which the change occurred. The average velocity is a vector quantity, meaning that it has both magnitude and direction.

The average velocity of an object gives a measure of its average speed and direction over a certain time interval. It is a useful concept in physics, especially in the study of mechanics and kinematics, where it is used to describe the motion of objects under different conditions and to calculate the velocity of objects at a specific instant in time.

The average velocity over an interval [a, b] is defined as the total change in position (y) divided by the total change in time (x), i.e., (y(b) - y(a)) / (b - a).

Given the function y = x² + 8x, we have:

y(5) = 5² + 8 × 5 = 25 + 40 = 65

y(8) = 8² + 8 × 8 = 64 + 64 = 128

So, the average velocity over the interval [5, 8] is:

(y(8) - y(5)) / (8 - 5) = (128 - 65) / (8 - 5) = 63 / 3 = 21.

Therefore, the average velocity of the function y = x² + 8x over the interval [5, 8] is 21.

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