A tracksuit costs £25. The price goes up by 5%. What is the new price?

Therefore, the requirements of Rolle's Theorem are not fulfilled, and the absence of a number c with f'(c) = 0 does not contradict the theorem.

To show that f(0) = f(π), we substitute the values into the function:

f(0) = tan(0) = 0

f(π) = tan(π) = 0

Hence, we have f(0) = f(π), indicating that the function values at x = 0 and x = π are equal.

To investigate the derivative, we differentiate f(x) = tan(x) with respect to x:

f'(x) = sec^2(x)

Next, we need to determine if there exists a number c in the interval (0, π) such that f'(c) = 0. Let's evaluate f'(x) at the endpoints of the interval:

f'(0) = sec^2(0) = 1

f'(π) = sec^2(π) = 1

Since f'(x) is always positive (1) for any x in the interval (0, π), there is no number c in that interval for which f'(c) = 0.

This observation does not contradict Rolle's Theorem because Rolle's Theorem requires three conditions to be satisfied:

The function must be continuous on the closed interval [a, b].

The function must be differentiable on the open interval (a, b).

The function values at the endpoints must be equal, i.e., f(a) = f(b).

In this case, f(x) = tan(x) fails to satisfy the second condition because the derivative, f'(x) = sec^2(x), is never zero in the interval (0, π).

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Answer:

72

Explanation:

First, we can find the top and bottom smaller squares of the rectangular prism. Since we are working with a variety of rectangles, we only need to use the equation L×W.

To start with, let's multiply 2×3, which gives us 6, the surface area of both the bottom and top rectangles, so now we need to multiply it by 2 to account for both of them. 6×2=12

Now, we'll find the surface area of the bigger rectangles in the middle, which are 6 by 3, so again we will need to multiply length times width, then by 2 to count both rectangles. 6×3=18×2=36

Finally, we can find the surface area of the smaller rectangles in the middle, which are 6 by 2. 6×2=12, then multiply by 2 since there are 2 of those rectangles, 12×2=24

Now to find the total surface area, we need to add the gathered surface area from each shape, 12+36+24=72

Answer:

14 and 4

Step-by-step explanation:

xy = 56

x - y = 10

x = 10 + y

(10+y)y = 56

10y + y² = 56

y² + 10y - 56 = 0

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

y - 4 = 0

y = 4

y + 14 = 0

y = -14

y is a positive integer so y = 4.

x - 4 = 10

x = 14

The two numbers whose product is 56 and difference is 10 are 14 and 4

A = 14

B = 4

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the first number be = A

Let the second number be = B

The product of two numbers is = A x B

The difference of two numbers is = A - B

Now , the equation is

A x B = 56

A - B = 10

Now ,

( A - B )² = A² + B² - 2 ( A x B )

Substituting the values of A and B in the equation , we get

100 = A² + B² - 112

Adding 112 on both sides , we get

A² + B²  = 212

When A = 14

B² = 212 - 196

B² = 16

So , B = 4

Therefore , the value of A is 14 and value of B is 4

Hence , The two numbers whose product is 56 and the difference is 10 are 14 and 4

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Answer:

if you are looking for the area, it would be 22ft

Step-by-step explanation:

Answer:

About 11.7

Step-by-step explanation:

In order to find the hypotenuse (if this is what you are looking for), you would use pythagorean theorem. This states : A squared + B squared = C squared. Our legs that you have given us (A and B) are 11 and 4. When we square these values, we get 121+16=137. Now we square the value of 137. To square the value, multiply numbers until you reach this number. Because the value 137 is odd, it will end up being a decimal when squared which is about 11.7

I also want to point out that all hypotenuse will be longer than the legs.

Answer:

There are 25 girls in the class.

Step-by-step explanation:

Number of girls=5x

Number of boys=3x

Number of students in the class= 5x+3x=8x=40

So, x= 40/8=5

Number of girls= 5x=5*5=25

Therefore, there are 25 girls in the class.

Due to the limited sample size and the uncertainty surrounding the coin's balance, the z test for a proportion is not appropriate in the scenario of tossing a coin 12 times to test the hypothesis that it is balanced.

The z test's presumptions could not hold true when the sample size is small (a). A substantial sample size is necessary for the z-test, which relies on the assumption that the sample has a normal distribution. The sample size is thought to be too small to satisfy this condition with only 12 coin tosses. As a result, using the z-test for proportions would not yield accurate findings.

The applicability of the z-test is further impacted by the uncertainty surrounding the coin's balance (b). In order to test a parameter (in this case, the proportion of heads or tails), the z-test presupposes that the null hypothesis is correct. We cannot, however, be assured that the coin is balanced in this circumstance.

The outcomes could be impacted by inherent biases or irregularities in the coin's design or tossing procedure. The z-test for proportions should not be used if the coin's balance is uncertain.

The z-test for proportions is therefore inappropriate in this situation due to both the tiny sample size and the ambiguity surrounding the coin's balance. For judging the fairness of the coin based on the provided sample, different statistical tests like the binomial test or the chi-square test would be more applicable.

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The Task objective chosen is: A Measuring Task

The Objective is: To investigate the fairness of a local athlete's track for runners starting on different lines.

To carry about the investigation, follow the steps given below.

Lastly, make a crossword puzzle that includes mathematical vocabulary in a clever manner.

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```Class IX```

*Mathematics Project work*

*Do any one*

1. *A Measuring Task*

To investigate your local athletes track to see whether it is worked fairly for runners who start on different lines.

2. *Design a crossword puzzle with mathematical terms*

To review mathematics vocabulary, to give the opportunity for creative expression in designing puzzles, to act as a means of motivating the study of a given unit and to given recreation .​

Answer:

3,4

Step-by-step explanation:

Answer:

-4

Step-by-step explanation:

To find slope, you find the rise and then the run. Then you put your answer in the format rise/run. If the line rises left to right, it's a positive slope and vice versa.

Hope it helps. :)

The series expansion for the derivative of  \(f\left(x\right)=\frac{1}{\:1-x^3}\) is

\(f'\left(x\right)=3x^2\:+6x^5+9x^8\:+\dots =\sum _{n=0}^{\infty }\:3n.x^{3n-1}\)

The given series expansion of \(f\left(x\right)=\frac{1}{\:1-x^3}\)

\(f(x)=1+x^3+x^6+x^9+----=\sum _{n=0}^{\infty }\:x^{3n}\)

To differentiate this series term-by-term, we need to find the derivative of each term.

Let's differentiate the individual term x³ⁿ.

\(\frac{d}{dx}\left(x^{3n}\right)=3n.x^{3n-1}\)

Now, we can express the derivative of f(x) as a series expansion:

\(f'\left(x\right)=\sum _{n=0}^{\infty }\:\frac{d}{dx}\left(x^{3n}\right)=\sum _{n=0}^{\infty }\:3n.x^{3n-1}\)

Therefore, the series expansion for the derivative of  \(f\left(x\right)=\frac{1}{\:1-x^3}\) is:

\(f'\left(x\right)=3x^2\:+6x^5+9x^8\:+\dots =\sum _{n=0}^{\infty }\:3n.x^{3n-1}\)

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The item that is not an item in the income statement is b. Furniture & Fixture as it is considered a fixed asset and is reported on the balance sheet instead.

The income statement, also known as the profit and loss statement, provides a summary of a company's revenues, expenses, gains, and losses over a specific period. It helps to assess the financial performance of a business. The income statement typically includes various items such as revenues, cost of goods sold, operating expenses, interest income or expense, and other gains or losses.

a. Discount allowed is a revenue item that represents the discounts given to customers as an incentive for early payment or other reasons. It is usually reported as a deduction from sales revenue.

c. Furniture & Fixture is not typically included in the income statement. Instead, it is considered a non-operating or non-recurring item and is generally classified as a fixed asset on the balance sheet. Fixed assets represent long-term investments made by a company for its operations.

d. Discount received is also not an item in the income statement. It represents the discounts received by a company from its suppliers for early payment or other reasons. Similar to discount allowed, it is usually reported as a deduction from the respective expense account.

In summary, b. Furniture & Fixture is the item that is not included in the income statement. It is considered a fixed asset and is reported on the balance sheet instead.

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Answer:

x=4 and x=0

Step-by-step explanation:

|x-2|=2

x-2=2 and

x-2= -2

x=4 and

x=0

Answer: X = |4|

|4-2|+10=12

Answer:

Step-by-step explanation:

Answer: 6y - 4

(-y + 5) + (7y - 9)

= (7y - y) + (5 - 9)

= 6y - 4

Step-by-step explanation:

Answer:

6y-4

Step-by-step explanation:

to solve this we are going to combine like terms using pemedas and other mathematical properties.

using the associative property, we know that we can rearrange the groupings of the parenthases

-y+5+7y-9

so now we are going to combine like terms

this leaves us with a final answer of  

6y-4    

The coordinate of the midpoint of line segment with endpoints (5,5) and (10,8) is ( 15/2, 13/2 ).

The midpoint formula is expressed as;

M = ( (x₁+x₂)/2, (y₁+y₂)/2 )

Given the data in the question;

Endpoint 1(5, 5)

Endpoint 2(10,8)

Plug the given points into the above formula and simplify.

M = ( (x₁+x₂)/2, (y₁+y₂)/2 )

M = ( ( 5 + 10 )/2, ( 5 + 8 )/2 )

M = ( 15 )/2, ( 13 )/2 )

M = ( 15/2, 13/2 )

Therefore, the midpoint of the segment is ( 15/2, 13/2 ).

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Answer:

she didnt go back from 15 she went back from 20and the arrows should be going backwords not forwards

The three errors that Emily makes are: The arrows should point in the negative direction. The arrows should end on negative 15. The arrows should start at zero.

The number line is a representation of numbers on the line which can be used for different arithmetic operations such as additions, subtractions, multiplication, and division of numbers and algebraic expressions. It usually starts from the origin 0 with the positive axis in the right direction and the negative axis in the left direction.

From the given information;

-15 ÷ 5 = -3

By representing the above information on the number line, the errors that Emily make with her diagram are:

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Answer:

The answer is C

Step-by-step explanation:

The roots of the quadratic equation are real and distinct

What is Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the quadratic equation be represented as A

Now , the value of A is

2x² + 4x - 7 = 0   be equation (1)

On simplifying the equation , we get

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

Substituting the values in the equation , we get

x = [ -4 ± √ ( ( 4 )² - 4 ( 2 ) ( -7 ) ) ] / 2 ( 2 )

On simplifying the equation , we get

x = [ -4 ± √ ( 16 + 56 ) ] / 4

x = [ -4 ± 6√2 ] / 4

x = ( -2 + 3√2 ) / 2

And ,

x = ( -2 + 3 ) / √2

Therefore , the roots are x = ( -2 + 3√2 ) / 2 and x = ( -2 + 3 ) / √2

Hence , the roots of the equations are real and distinct

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Answer:

6

Step-by-step explanation:

24 divided by 4 is 6

(a) The figure is line Segment

(b)The figure is line.

(c )The figure is ray.

Line segment: An part of a line with two endpoints is called a line segment. Different shapes, including triangles, polygons, hexagons, and squares, are constructed using a variety of line segments. A line segment's length is its measurement.

Line: A line has length but no width, making it a one-dimensional figure. A line is made up of a collection of points that can be stretched indefinitely in opposing directions.

Ray: Ray is a further link in the chain. It consists of a line and a line segment combination with one terminating end and an eternally extending end. Its length cannot be calculated since one of its ends is non-terminating.

Line segment, Line and Ray:

A line segment is a section of a line with two ends that has a quantifiable length. A ray, on the other hand, has only one fixed point (its starting point), but no endpoint. Therefore, its length cannot be determined.A line can be extended on both sides and has no ends. In contrast, a line segment cannot be extended on both sides because it has two endpoints.

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The steps to solve a quadratic formula include:

In algebra, a quadratic equation is a equation that can be rearranged in standard form as where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0.

A quadratic equation is an equation that could be written as ax² + bx + c = 0

when a 0. The factoring method is illustrated:

To solve a quadratic equation by factoring, put all terms on one side of the equal sign, leaving zero on the other side.

Set each factor equal to zero.

Solve each of these equations.

Check by inserting your answer in the original equation.

Example 1

Solve x² – 6 x = 16.

Following the steps,

x² – 6 x = 16 becomes x² – 8x + 2x – 16 = 0

Factor.

( x – 8)( x + 2) = 0

Therefore, x = 8 and -2.

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Restaurant A offers the best value as one hamburger is $1.29

The first step is to calculate the price of one burger in restaurant A

11= 14.19

1= x

cross multiply

11x= 14.19

x= 14.19/11

x= 1.29

The next step is to calculate the price on one burger in restaurant B

2= 2.64

1= x

Cross multiply

2x= 2.64

x= 2.64/2

x= 1.32

Hence the price of the hamburger at Restaurant A is $1.29 and the price at Restaurant B is $1.32.

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Answer:

(A) Restaurant A for $1.29

Step-by-step explanation:

Got it right on my quiz.

Approximately 35% of the ears of corn in the bin are white.

To find the percentage of white ears of corn, we need to determine the number of white ears and then calculate what proportion they make out of the total number of ears in the bin.

Given that there are 78 ears of yellow corn, we can subtract this number from the total number of ears to find the number of white ears:

Total ears - Yellow ears = White ears

120 - 78 = 42

There are 42 white ears of corn in the bin. To calculate the percentage of white ears, we divide the number of white ears by the total number of ears and multiply by 100:

(White ears / Total ears) x 100 = Percentage of white ears

\((42 / 120) \times 100 = 35%\)

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The main advantage of using a graphic calculator is that it speeds up the process of solving more difficult arithmetic problems for students.

A graphing calculator, often known as a graphics calculator or graphic display calculator, is a portable computer that can plot graphs, solve many equations at once, and carry out other variable-related activities. The most common graphing calculators are programmable calculators, which let the user design unique programs, generally for scientific, technical, or educational purposes. They feature big screens that show computations and several lines of text.

Numerous graphing calculators may be connected to sensors such as accelerometers, electronic thermometers, pH meters, weather stations, decibels, light meters, and other devices to operate as data recorders and as monitoring, polling, and communication tools with teachers. The use of such data in student laboratory tasks improves their understanding of math, particularly statistics and mechanics. 

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The Mean Absolute Error (MAE) is [Insert value]. The Mean Squared Error (MSE) is [Insert value].

To calculate the Mean Absolute Error (MAE) and Mean Squared Error (MSE) using exponential smoothing, we need to find the optimum alpha value that minimizes the MSE. Excel Solver can be used to determine this value.

Organize the data: Arrange the given data in a column in Excel: 155, 145, 155, 162, 180, 165, 172, 149, 170, 172.

Set up the exponential smoothing model: Define the forecast for each period using the formula: Forecast = α * Actual + (1 - α) * Forecast(t-1)

Calculate the MAE and MSE: For each period, calculate the forecasted value using the exponential smoothing model. Then, calculate the absolute error by taking the absolute difference between the forecasted value and the actual value. Square each absolute error to obtain the squared error. Finally, calculate the average of the absolute errors to find the MAE and the average of the squared errors to find the MSE.

Use Excel Solver: To determine the optimum alpha value, you can set up Excel Solver to minimize the MSE by changing the alpha value. The objective is to find the alpha value that gives the lowest MSE.

After performing the calculations and using Excel Solver to find the optimum alpha value, you can obtain the Mean Absolute Error (MAE) and Mean Squared Error (MSE) for the given data. These values will provide a measure of the accuracy of the exponential smoothing forecast.

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Answer:

3750

Step-by-step explanation:

25,000 = 100%

20% accomodation

50% household

15% transportation

Saved=100%-20%-50%-15%=15% left saved

15% of 25,000=3750

_______________________________

_______________________________

(a) tangent line to the graph of f(x) = x^3 at the point (4,2).

(b) equation of the tangent line to the graph of f(x) = x^2 - x at the point (4,12).

(a) To find the equation of the tangent line to the graph of f(x) = x^3 at the point (4,2), we need to find the slope of the tangent line at that point. We can do this by taking the derivative of f(x) with respect to x and evaluating it at x = 4. The derivative of f(x) = x^3 is f'(x) = 3x^2. Evaluating f'(x) at x = 4 gives us the slope of the tangent line. Once we have the slope, we can use the point-slope form of a linear equation to write the equation of the tangent line.

(b) Similarly, to find the equation of the tangent line to the graph of f(x) = x^2 - x at the point (4,12), we differentiate f(x) to find the derivative f'(x). The derivative of f(x) = x^2 - x is f'(x) = 2x - 1. Evaluating f'(x) at x = 4 gives us the slope of the tangent line. Using the point-slope form, we can write the equation of the tangent line.

In both cases, the equations of the tangent lines will be in the form y = mx + b, where m is the slope and b is the y-intercept.

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The common difference between each term is -12.

To solve this problem, we can use the formula for the sum of an arithmetic series:

Sn = n/2 [2a + (n-1)d]

where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.

Using this formula, we can write:

T₁ + T₂ + T₃ + ... + T2013 = 2013/2 [2T₁ + (2013-1)d]

T₂ + T₁ + T₂ + T₃ + ... + T2014 = 2014/2 [2T₁ + (2014-1)d]

Since T₂ + T₁ + T₂ + T₃ + ... + T2014 = 6050, we can substitute this expression into the second equation and simplify:

6050 = 1007 [2T₁ + 2013d]

Dividing both sides by 1007, we get:

6 = 2T₁/1007 + d

We also know that T₁ + T₂ + T₃ + ... + T2013 = 3029, so we can substitute this expression into the first equation and simplify:

3029 = 1006 [2T₁ + 2012d]

Dividing both sides by 1006, we get:

3 = T₁/503 + d

Now we have two equations with two unknowns (T₁ and d). We can solve for d by subtracting the second equation from the first:

6 - 3 = 2T₁/1007 + d - (T₁/503 + d)

Simplifying, we get:

3 = T₁/1007

Multiplying both sides by 1007, we get:

T₁ = 3021

Now we can substitute T₁ into one of the earlier equations to solve for d:

3 = T₁/503 + d

3 = 3021/503 + d

d = -12

Therefore, the common difference between each term is -12.

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Answer:

\(\frac{5}{1+5}\)

Step-by-step explanation:

Hope this helps :))

Answer:

0.4

Step-by-step explanation:

4 juniors out of a total of 10

4/10 = 0.4