Answer:
Representation B does not show the amount of time Kaleigh spent watching TV at this rate. Representation B only shows the total time she would have spent based on the number of episodes watched, assuming each episode is 30 minutes long. It does not take into account the actual time it took for Kaleigh to watch the episodes, which may have varied depending on how quickly she watched them.
Step-by-step explanation:
The volume of a cylinder is 12,566.4 cm cubed. The height of the cylinder is 8 cm.
a) Find the radius of the cylinder to the nearest tenth of a centimeter.
b) Find the area of the base of the cylinder to the nearest tenth of a centimeter.
c) Find the lateral area to the nearest tenth of a centimeter.
d) Find the surface area of the cylinder.
As a result of the cylinder's volume being 12,566.4 centimetres³ and its height being 8 cm, the surface area of the cylinder is approximately 942.5 cm².
what is volume ?
Volume in mathematics refers to how much room a three-dimensional object takes up. It is a way to quantify how much enclosed room there is overall in a solid figure. Depending on the shape of the item, a number of formulas can be used to calculate its volume. A rectangular prism's volume, for instance, can be calculated by multiplying its length, breadth, and height, whereas a cylinder's volume can be calculated by dividing its base area by its height. The system being used will determine the unit of measurement for volume, but typical units include cubic metres (m³) and cubic centimetres (cm³).
given
a) V = πr²h, where V is the volume, r is the radius, and h is the height, is the expression for a cylinder's volume. By rearranging the calculation, we can find r:
V/(h) = √12,566.4/(*8)) Equals r ≈ 10 centimetres (rounded to the closest tenth of a cm) (rounded to the nearest tenth of a cm)
Consequently, the cylinder's radius is roughly 10 centimetres.
b) The equation A = r² determines the area of a cylinder's base. Using the r number we just discovered, we have:
A = π(10)² ≈ 314.2 cm² (rounded to the closest tenth of a cm) (rounded to the nearest tenth of a cm)
Consequently, the cylinder's base has a surface area of about 314.2 cm².
c) The equation L = 2πrh, where r is the radius and h is the height, determines the lateral area of a cylinder. With the numbers from the problem substituted, we obtain:
L = 2π(10)(8) ≈ 502.7 cm² (rounded to the closest tenth of a cm) (rounded to the nearest tenth of a cm)
Consequently, the cylinder's side area is roughly 502.7 cm².
d) The equation S = 2πr² + 2πrh gives the surface area of a cylindrical. Using the r and h numbers we discovered earlier, we have:
S = 2π(10)² + 2π(10)(8) ≈ 942.5 cm² (rounded to the nearest tenth of a centimetre) (rounded to the nearest tenth of a cm)
As a result of the cylinder's volume being 12,566.4 centimetres³ and its height being 8 cm, the surface area of the cylinder is approximately 942.5 cm².
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Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options. x < 5 –6x – 5 < 10 – x –6x + 15 < 10 – 5x A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right. A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
The correct representations of the inequality –3(2x – 5) < 5(2 – x) are given as follows:
x > 5.An open circle is at 5 and a bold line starts at 5 and is pointing to the right.What are the inequality symbols?The four inequality symbols, along with their meaning, are presented as follows:
> x: amount greater than x -> to the right of x with an open dot at the number line. -> above the dashed horizontal line y = x on the coordinate plane.< x: amount less than x. -> to the left of x with an open dot at the number line. -> below the dashed horizontal line y = x on the coordinate plane.≥ x: amount at least x. -> to the right of x with a closed dot at the number line. -> above the solid horizontal line y = x on the coordinate plane.≤ amount at most x. -> to the left of x with a closed dot at the number line. -> above the dashed horizontal line y = x on the coordinate plane.The inequality for this problem is defined as follows:
–3(2x – 5) < 5(2 – x)
Expanding the operations, the solution is obtained as follows:
-6x + 15 < 10 - 5x
-6x + 5x < 10 - 15
-x < -5
x > 5.
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Find the volume of the figure below in terms of pi.
A. 36pi
B: 288pi
C: 144pi
D: 864pi
Work Shown:
[tex]V = \text{Volume of a sphere}\\\\V = \frac{4}{3}\pi*r^3\\\\V = \frac{4}{3}\pi*6^3\\\\V = \frac{4}{3}\pi*216\\\\V = \frac{4}{3}*216\pi\\\\V = 288\pi\\\\[/tex]
-15y-4
45
27
38%
十
15
24
Find X and
Find Y
Answer:
42
Step-by-step explanation:
you asding it all up
1 Select the correct answer. What is the simplified form of this expression? (-3x² + 4x) + (2x²-x-11)
a. -x2 + 5x − 11
b. -x² + 3x - 11
c. -x² + 3x + 1
d. -x2 + 5x + 11
Blynomials: Mastery Test
The value of the given expression is - x² + 3x - 11 and option b is the correct answer.
What is binomial expression?Binomial is the name for an algebraic expression with only two terms. It is a polynomial with two terms. It is sometimes referred to as the sum or difference of two or more monomials. It is a polynomial's most basic form. Therefore, A binomial is a two-term algebraic statement that includes a constant, exponents, a variable, and a coefficient.
The given expression is:
(-3x² + 4x) + (2x²-x-11)
-3x² + 4x + 2x² - x - 11
Subtract the like terms:
- x² + 3x - 11
Hence, the value of the given expression is - x² + 3x - 11 and option b is the correct answer.
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Find the total cost of tiling a triangular area having a base
length of 3 meters and a height of 9 meters if it costs $9.71 to
tile one square meter. Round to the nearest cent
The total cost of tiling a triangular area having a baselength of 3 meters and a height of 9 meters is $131.29.
Given base length of triangular area is 3 metersHeight of triangular area is 9 metersCost of tiling one square meter = $9.71 We need to find the total cost of tiling the triangular area. The area of a triangle is given by the formula as shown below,
Area of a triangle = 1/2 × base length × height, Therefore, Area of the given triangle = 1/2 × 3 meters × 9 meters= 13.5 square meters. Now, the total cost of tiling the triangular area can be found by multiplying the area by the cost per square meter. Total cost of tiling triangular area = 13.5 square meters × $9.71/square meter = $131.29 (approx)
Hence, the total cost of tiling a triangular area having a baselength of 3 meters and a height of 9 meters is $131.29.
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DUE TODAY PLEASE HELP NOW!!!!!!!!!!!!!
Here is another triangle similar to DEF found in the lesson section labeled “Shrinking Triangles”.
• Label the triangle D”E”F”.
• What is the scale factor from triangle DEF to triangle D”E”F”?
• What are the coordinates of F”? Explain how you know.
• What are cos(D”), sin(D”), and tan(D”)?
The scale factοr οf dilatiοn is 1/40 and the cοοrdinates οf F" are (1440, 600)
Labelling the image οf the triangle
The image οf the label is attached
The scale factοr οf the dilatiοn
Given that
D'E' = 36 units
Then, we have
D"E" = 0.9 units
The scale factοr is calculated as
Scale factοr = D"E"/D'E'
Sο, we have
Scale factοr = 0.9/36
Evaluate
Scale factοr = 1/40
Sο, the scale factοr οf dilatiοn is 1/40
The cοοrdinates οf F"
This is calculated as
F = F'/Scale factοr
Sο, we have
F = (36, 15)/(1/40)
F = (1440, 600)
The trigοnοmetry ratiοs
The trigοnοmetry ratiοs are calculated as
sin(D") = EF/DF
cοs(D") = DE/DF
tan(D") = EF/DE
Sο, we have the fοllοwing apprοximated values
sin(D") = 15/39 = 0.36
cοs(D") = 36/39 = 0.92
tan(D") = 0.36/0.92 = 0.39
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the population in a particular country is growing at the rate of 1.6% per year. If 5,092,000 people lived there in 1999, how many will there be in the year 2003? Use f(x)=y 0^e^0.016t and round to the nearest ten-thousand.
The population in 2003 is 5,830,000.
The population in a particular country is growing at the rate of 1.6% per year. To calculate the population for 2003, we can use the formula f(x)=y[tex]0^e^0.016t[/tex] .
Where y0 is the population in 1999 and t is the time in years.
Plugging in the numbers we have y0 = 5092000 and t = 4 for the year 2003, the formula looks like this: f(x)=5092000e0[tex]^{016(4)[/tex].
Solving this equation, we get f(x)=5837280, which is the population in 2003. Rounding this number to the nearest ten-thousand, the population in 2003 is 5,830,000.
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Talia deposits $350 in her savings account. The account earns 2.5% simple interest per year. What is the balance in the account after 4 years?
Answer:385
Formulate and substitute:F=350+350rt
Calculate the product or quotient:350x1+0.1
Calculate the sum or difference:350x1.1
Calculate the product or quotient:385
The height of a plant over time is shown in the table below. Using a logarithmic model, what is the best estimate for the age of the plant when it is 19 inches tall?
Moreover, we should evaluate the model's goodness of fit and take into expressions account additional elements that can influence plant development.
what is expression ?Mathematical operations include doubling, dividing, adding, and subtracting. A phrase is constructed as follows: Expression, monetary value, and mathematical operation Numbers, parameters, and functions make up a mathematical expression. It is possible to use words and terms in contrast. Every mathematical statement including variables, numbers, and a mathematical operation between them is called an expression, sometimes referred to as an algebraic expression. For example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.
In the formula, h is the height, t is the time, and a and b are the parameters that need to be approximated.
The information in the table may be used to estimate the values of a and b. The equation is first transformed to yield:
[tex]log(b*t) = log(h/a)[/tex]
[tex]19 = 19.24 * log(53.89*t)[/tex]
t = 0.186 months, or approximately 5.58 days, or log(53.89*t) = 1 53.89*t = 10
Hence, 5.58 days is the best guess for the age of the plant at 19 inches tall. Seeing that this is a very little period of time, it is probable that the model will not be correct for values of t this low. Moreover, we should evaluate the model's goodness of fit and take into account additional elements that can influence plant development.
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What are the solutions of the equation 0=x^2-9x+8
The solutions of the equation 0=x^2-9x+8 are x = 8 and x = 1
How to determine the solutions of the equationFrom the question, we have the following parameters that can be used in our computation:
0=x^2-9x+8
Rewrite as
x^2 - 9x + 8 = 0
Expand the equation
So, we have the following representation
x^2 - 8x - x + 8 = 0
When the equation is factorized, we have
(x - 8)(x - 1) = 0
This gives
x - 8 = 0 and x - 1 = 0
Evaluate
x = 8 and x = 1
Hence, the solutions are x = 8 and x = 1
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Identify the following models as ARMA(p, q) models (watch out for parameter redundancy), and determine whether they are causal and/or invertible:
(a) xt =. 80xt-1 −. 15xt−2 + wt −. 30wt−1.
(b) xt = xt−1 −. 50xt−2 + wt − wt−1.
The first model (a) is an ARMA(1,2) model and is both causal and invertible, while the second model (b) is an ARMA(2,1) model and is also both causal and invertible.
What are Autoregressive (AR) models?Autoregressive (AR) models are a type of statistical model that uses the past values of a variable to predict its future values. The main assumption of AR models is that the future values of the variable are correlated with its past values. They are used to analyze time series data and are commonly used in forecasting, price and demand prediction, and other time-dependent processes.
The first model (a) is an ARMA(1,2) model, which is a combination of an autoregressive (AR) and a moving average (MA) model. This model is both causal and invertible. Causal models are models that have a lag of only one period, while invertible models are those that have a lag of two or more periods. The second model (b) is an ARMA(2,1) model, which is a combination of an autoregressive (AR) and a moving average (MA) model. This model is both causal and invertible. Since the lags for both models are greater than one, both models are invertible.
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theres a picture of what I need help with thank you so much! Have an amazing day and will also mark brainliest <3
Answer:
5x4
Step-by-step explanation:
The dimensions of the rectangle are 20x16
The reducing factor is 0.25.
Multiple 20 * 0.25= 5
Multiple 16 * 0.25= 4
Therefore the new dimensions are 5x4.
If a+x=b+y=c+z+1, where a,b,c,x,y, z are non-zero distinct real numbers. Then [[x,a+y,x+a],[y,b+y,y+b],[z,c+y,z+c]]
The value of the matrix is [[b+y-a,a+x-b,c+z+1-a],[a+x-b,b+y-a,c+z+1-b],[a+x-c-1,b+y-c-1,c+z]] when a+x=b+y=c+z+1, where a,b,c,x,y, z are non-zero distinct real numbers.
According to the given equation, a+x=b+y=c+z+1, where a,b,c,x,y, z are non-zero distinct real numbers, we can rearrange the equation to find the value of one of the variables in terms of the others. For example, we can rearrange the equation to find the value of x in terms of the other variables:
x = b+y-a = c+z+1-a
Similarly, we can rearrange the equation to find the value of y and z in terms of the other variables:
y = a+x-b = c+z+1-b
z = a+x-c-1 = b+y-c-1
Now, we can substitute these values into the given matrix to find the value of each element:
[[x,a+y,x+a],[y,b+y,y+b],[z,c+y,z+c]] = [[b+y-a,a+x-b,c+z+1-a],[a+x-b,b+y-a,c+z+1-b],[a+x-c-1,b+y-c-1,c+z+1-c-1]]
Simplifying the matrix, we get:
[[b+y-a,a+x-b,c+z+1-a],[a+x-b,b+y-a,c+z+1-b],[a+x-c-1,b+y-c-1,c+z]]
Therefore, the value of the matrix is [[b+y-a,a+x-b,c+z+1-a],[a+x-b,b+y-a,c+z+1-b],[a+x-c-1,b+y-c-1,c+z]] when a+x=b+y=c+z+1, where a,b,c,x,y, z are non-zero distinct real numbers.
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Please help me answer this
Answer: I think B & C not super sure though
Select the graph of the solution set that would represent the following expression. 3(x - 2) = 5(x + 1)
Answer:
Step-by-step explanation:
To graph the equation 3(x - 2) = 5(x + 1), we can first simplify it by expanding the brackets:
3x - 6 = 5x + 5
Next, we can isolate the variable on one side of the equation. We can do this by subtracting 3x from both sides and adding 6 to both sides:
-11 = 2x
x = -11/2
Now we have the x-coordinate of the point where the graph of the equation intersects the x-axis. To find the y-coordinate of this point, we can substitute x = -11/2 into one of the original equations and solve for y:
3(x - 2) = 5(x + 1)
3(-11/2 - 2) = 5(-11/2 + 1)
-33/2 - 6 = -55/2 + 5
-33/2 - 6 + 55/2 = 5
-33/2 + 44/2 = 5
11/2 = 5
Abbie bought 82 cases of water for her restaurant Each case had 24 bottles of water How many bottle of water did Abbie buy in all?
Answer:
1968 bottles of water
Step-by-step explanation:
24×82=1968
1968 bottles of water
A company makes a single serving cereal box that contains a volume of 1. 2 ounces of cereal. They plan to make an extra large box for school cafeterias. The extra large box will be dilation of the single serving box using a scale factor of 4. How many ounces of cereal will the extra large box contain?
If the extra large box have a dilation scale factor of 4, then the extra large box contain 76.8ounces.
If the extra large box is a dilation of the single serving box with a scale factor of 4, then
The ratio of the volumes of the extra large box to the single serving box will be 4³,
We know that, Volume is a three-dimensional measurement and is affected by the cube of the scale factor.
So, the volume of the extra large box will be;
⇒ 4³ × 1.2 ounces = 4×4×4×1.2 ounces = 76.8 ounces,
Therefore, the extra large box will contain 76.8 ounces of cereal.
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Determine the x-intercepts of each of the following functions. 2. y=x^2+5x−24 3. y=x^2−11x+10
The x-intercepts of y=x2+5x−24 can be determined by solving the equation 0=x2+5x−24. Using the quadratic formula, we can find that x = 6 and x = -4 are the two x-intercepts.
The x-intercepts of y=x2−11x+10 can be determined by solving the equation 0=x2−11x+10. Using the quadratic formula, we can find that x = 5 and x = -2 are the two x-intercepts.
To determine the x-intercepts of the given functions, we need to find the values of x that make y equal to 0. We can do this by factoring the equations and setting each factor equal to 0.
2. y = x^2 + 5x - 24
0 = x^2 + 5x - 24
0 = (x + 8)(x - 3)
x + 8 = 0 or x - 3 = 0
x = -8 or x = 3
The x-intercepts are (-8, 0) and (3, 0).
3. y = x^2 - 11x + 10
0 = x^2 - 11x + 10
0 = (x - 1)(x - 10)
x - 1 = 0 or x - 10 = 0
x = 1 or x = 10
The x-intercepts are (1, 0) and (10, 0).
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We measure three dimensional space with volume. Volume is how much three dimensional space something takes up. We usually measure it in cubic meters or cubic feet. 90 ft
We usually measure it in cubic meters or cubic feet. 90 ft³ is equal to 2,544.48 cm³.
What is length?Length is a term used to describe the magnitude of a line, distance, or size. It usually refers to the measurement of something from end to end, such as the length of a river, the height of a building, or the size of a piece of fabric. Length is typically measured in units such as feet, meters, or even inches. It is an important concept in science, engineering, and mathematics. Length helps us to understand and quantify the size of objects and the space they occupy.
We can find the volume of any three dimensional object by using the formula V = l x w x h, where V is the volume, l is the length of the object, w is the width, and h is the height. This formula is applicable to all three dimensional shapes, including cubes, rectangles, prisms, cylinders, and so on.
To find the volume of a cylinder, we use the formula V = π x r² x h. Here, V is the volume, π is the constant pi, r is the radius of the cylinder, and h is the height. To find the volume of a sphere, we use the formula V = 4/3 x π x r³. Here, V is the volume, π is the constant pi, and r is the radius of the sphere.
We can also use volume to measure liquids or gases. We measure liquids in liters or milliliters, and gases in cubic meters or cubic feet. For example, 1 liter of water is the same as 1,000 milliliters of water.
Volume is an important concept in mathematics and is used to measure many different things. It is used to measure the size of objects, the volume of liquids and gases, and the amount of three dimensional space something occupies. Understanding volume and how to calculate it is an essential skill for anyone studying mathematics.
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what is the quadratic formula multiplied by pie divided by X?
The quadratic formula multiplied by π divided by x is: π(-b ± √(b² - 4ac)) / (2ax).
What is the Quadratic formula?
The quadratic formula is used to solve quadratic equations, and is given by: x = (-b ± √(b² - 4ac)) / 2a.
Pi otherwise denoted by the symbol π in real terms is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. It is approximately equal to 3.14159, although its decimal representation goes on infinitely without repeating.
Since the rational objective of every mathematical expression or problem is to simplify, we will leave Pi in it's symbolic form - π.
Thus, multiplying quadratic formula by π and dividing by x, we get:
π(-b ± √(b² - 4ac)) / (2ax)
Therefore, the quadratic formula multiplied by π divided by x is:
π(-b ± √(b² - 4ac)) / (2ax).
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Answer: π(-b ± √(b² - 4ac)) / (2ax)
Step-by-step explanation:
take the standard quadratic formula and plug in pi and x and you get you answer hope this helps
2. (4 marks) Find a basis for \( \mathbb{R}^{4} \) containing \( v=(1,-1,1,-1), \quad w=(0,1,0,1) \).
A basis for \( \mathbb{R}^{4} \) containing \( v \) and \( w \) is \( \{v, w, e_{1}, e_{2}\} = \{(1,-1,1,-1), (0,1,0,1), (1,0,0,0), (0,1,0,0)\} \).
A basis for \( \mathbb{R}^{4} \) is a set of four linearly independent vectors that span \( \mathbb{R}^{4} \). We are given two vectors \( v=(1,-1,1,-1) \) and \( w=(0,1,0,1) \) that are part of the basis. To find the other two vectors, we can use the standard basis vectors \( e_{1}=(1,0,0,0) \) and \( e_{2}=(0,1,0,0) \) and check if they are linearly independent with \( v \) and \( w \).
First, we check if \( e_{1} \) is linearly independent with \( v \) and \( w \). We can do this by setting up the equation \( a_{1}v + a_{2}w + a_{3}e_{1} = 0 \) and solving for the coefficients \( a_{1}, a_{2}, \) and \( a_{3} \).
\( a_{1}(1,-1,1,-1) + a_{2}(0,1,0,1) + a_{3}(1,0,0,0) = (0,0,0,0) \)
This gives us the system of equations:
\( a_{1} + a_{3} = 0 \)
\( -a_{1} + a_{2} = 0 \)
\( a_{1} = 0 \)
\( -a_{1} + a_{2} = 0 \)
We can see that the only solution is \( a_{1}=a_{2}=a_{3}=0 \), which means that \( e_{1} \) is linearly independent with \( v \) and \( w \).
Next, we check if \( e_{2} \) is linearly independent with \( v \), \( w \), and \( e_{1} \) by setting up the equation \( a_{1}v + a_{2}w + a_{3}e_{1} + a_{4}e_{2} = 0 \) and solving for the coefficients \( a_{1}, a_{2}, a_{3}, \) and \( a_{4} \).
\( a_{1}(1,-1,1,-1) + a_{2}(0,1,0,1) + a_{3}(1,0,0,0) + a_{4}(0,1,0,0) = (0,0,0,0) \)
This gives us the system of equations:
\( a_{1} + a_{3} = 0 \)
\( -a_{1} + a_{2} + a_{4} = 0 \)
\( a_{1} = 0 \)
\( -a_{1} + a_{2} = 0 \)
Again, we can see that the only solution is \( a_{1}=a_{2}=a_{3}=a_{4}=0 \), which means that \( e_{2} \) is linearly independent with \( v \), \( w \), and \( e_{1} \).
Therefore, a basis for \( \mathbb{R}^{4} \) containing \( v \) and \( w \) is \( \{v, w, e_{1}, e_{2}\} = \{(1,-1,1,-1), (0,1,0,1), (1,0,0,0), (0,1,0,0)\} \).
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sharah has 35 coins in her pocket, all of which are dimes and quarters. if they are worth a total of $5.30, how many of each kind dose she have?
The number of each kind of coins Sharah have is 23 dimes and 12 quarters.
Let's use a system of equations to solve this problem. Let x be the number of dimes and y be the number of quarters that Sharah has. Then, we can write the following equations:
x + y = 35 (the total number of coins)
0.10x + 0.25y = 5.30 (the total value of the coins)
We can use the first equation to solve for one of the variables in terms of the other. Let's solve for x:
x = 35 - y
Now we can substitute this into the second equation:
0.10(35 - y) + 0.25y = 5.30
3.50 - 0.10y + 0.25y = 5.30
0.15y = 1.80
y = 12
Now we can use this value of y to solve for x:
x = 35 - 12 = 23
So Sharah has 23 dimes and 12 quarters.
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I put a photo please help
The requried combined length of both trains is 900 meters or 900 km.
What is speed?Speed is defined as when an object is in motion, the distance covered by that object per unit of time is called speed.
Here,
Let's first convert the given speeds from km/h to m/s to make the calculations easier.
Speed of the first train = 102 km/h = (102 x 1000) m/3600 s = 28.33 m/s
Speed of the second train = 30 km/h = (30 x 1000) m/3600 s = 8.33 m/s
Now, let's consider the relative speed of the two trains since they are moving in the same direction:
Relative speed = Speed of first train - Speed of second train
= 28.33 - 8.33
= 20 m/s
Let's assume that the length of the first train is x m and the length of the second train is y m.
Distance covered by first train in 45 seconds = (x + y) m
Speed of first train = Distance covered by first train / Time taken
= (x + y) / 45 km/s
We know that the speed of the first train is 20 m/s. So, we can write:
20 = [(x + y) / 45]
Solving for (x + y),
x + y = 900 m
Therefore, the combined length of both trains is 900 meters or 900 km.
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given f(x)=x^3-27x+5 answer the following:
Is the function increasing or decreasing at x=2?
List the interval (a,b) where the f(x) is increasing.
At what x-value does f(x) have a relative maximum?
(a) The function is decreasing at x=2.
(b) The interval where f(x) is increasing is (-3, 3).
(c) The function, f(x) has a relative maximum at x=-3.
Is the function increasing or decreasing at x=2?
To determine whether the function is increasing or decreasing at x=2, we need to find the derivative of f(x) and evaluate it at x=2.
f(x) = x³ - 27x + 5
f'(x) = 3x² - 27
At x=2, f'(2) = 3(2)² - 27 = -15, which is negative.
Therefore, the function is decreasing at x=2.
To find the interval where f(x) is increasing, we need to find the values of x where f'(x) > 0.
f'(x) = 3x² - 27
3x² - 27 > 0
3(x² - 9) > 0
(x - 3)(x + 3) > 0
The critical points are x=-3 and x=3.
We can test each of the intervals created by these critical points to see where f(x) is increasing or decreasing.
When x < -3, f'(x) < 0, so f(x) is decreasing.
When -3 < x < 3, f'(x) > 0, so f(x) is increasing.
When x > 3, f'(x) > 0, so f(x) is increasing.
Therefore, the interval where f(x) is increasing is (-3, 3).
To find the relative maximum of f(x), we need to find the critical points where f'(x) = 0.
3x² - 27 = 0
x² - 9 = 0
(x - 3)(x + 3) = 0
The critical points are x=-3 and x=3.
To determine which of these critical points corresponds to a relative maximum, we need to use the second derivative test.
f''(x) = 6x
At x=-3, f''(-3) = -18, which is negative.
Therefore, f(x) has a relative maximum at x=-3.
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O is the center of the regular octagon below. Find its perimeter. Round to the nearest tenth if necessary.
The perimeter of the regular octagon with an apothem of 5 units will be 33.14 units.
What is the perimeter of the regular polygon?All the sides of the regular polygon are congruent to each other. The perimeter of the regular polygon of n sides will be the product of the number of the side and the side length of the regular polygon.
P = (Side length) x n
The Apothem of a regular octagon is 5 units. Then the side length of the regular octagon is given as,
tan (360° / (2 × 8)) = (n/2) ÷ 5
tan 22.5° = n / 10
n = 4.142
Then the perimeter is given as,
P = 8 x 4.142
P = 33.14 units
The perimeter of the regular octagon with an apothem of 5 units will be 33.14 units.
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Which system of inequalities is represented by the graph?
The following system of inequalities is represented by the graph
x+y>3
-x+y< -4
What is inequality?
An inequality in mathematics depicts the connection between two values in an algebraic statement that are not equal. One of the two variables on the two sides of the inequality may be greater than, greater than or equal to, less than, or less than or equal to another value, according to inequality signals.
Consider the inequalities:
x+y>3-------(1)
-x+y< -4------(2)
Find x and y intercepts of both by considering them as equations by using equal sign in the place of inequality sign
x+y=3-------(3)
-x+y= -4------(4)
Find x and intercepts of these 2 equations:
we get x intercept when y=0 ( substitute y=0 into the respective equations and find x)
we get y intercept when x=0 ( substitute x=0 into the respective equations and find y)
x intercept of (3) is (3,0)
y intercept of (3) is (0,3)
x intercept of (3) is (4,0)
y intercept of (3) is (0,-4)
plot these points on a the plane and join them with lines
use dotted line and shade under the lines because of less than symbol
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What is the largest possible value x could take given that it must be an integer? x < 2
The largest possible value x could take given that it must be an integer for x < 2 is 1.
Difference between an inequality and an equation?The key distinction between an inequality and an equation is that an inequality describes a connection of inequality between two expressions, whereas an equation expresses equality between two expressions. In other words, an inequality shows that one expression is more or less than the other expression, but an equation shows that two expressions have the same value.
The highest number x might have is 1 if x must be an integer and x 2. This is due to the fact that any number higher than one would break the inequality x 2, and any number between one and two that is not an integer would also violate the stipulation that x must be an integer.
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Use the Intermediate Value Theorem to show thatf has a zero
between a and b .
f(x)=2x 3 +6x 2
-3 a -2
There is a zero for f(x) between a and b.
How to calculateUsing the Intermediate Value Theorem, we can show that the function f(x)=2x3+6x2-3a-2 has a zero between a and b.
The Intermediate Value Theorem states that if a continuous function f(x) takes on different values at two points a and b, then there must exist at least one value c in between a and b such that f(c) = 0. Since f(x) is a continuous function, it takes on different values at a and b.
Thus, by the Intermediate Value Theorem, there must exist at least one value c in between a and b such that f(c) = 0, meaning that there is a zero for f(x) between a and b.
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Question 4 Solve the inequality and write your answer in interval notation 16+12x<19x+12
The solution to the inequality 16 + 12x < 19x + 12 is all the values of x between the interval notation (4/7, ∞).
To solve the inequality 16 + 12x < 19x + 12, we need to isolate the variable x on one side of the inequality. We can do this by subtracting 12x from both sides and subtracting 12 from both sides:
16 + 12x - 12x - 12 < 19x + 12 - 12x - 12
4 < 7x
Solving for x, we can divide both sides by 7:
4/7 < x or x > 4/7
In interval notation, this would be (4/7, ∞). Therefore, the solution to the inequality is (4/7, ∞) in interval notation.
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