Mathmatics for Calculus

Among 60 students the probability that the group plays Field Hockey (F) and Basketball (B) is 0.083.

What is probability?

Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can range from 0 to 1, with 0 denoting an impossibility and 1 denoting a certainty.

From the given Venn Diagram -

The number of students who play Field Hockey (F) is = 18

The number of students who play Soccer (S) is = 10

The number of students who play Basketball (B) is = 7

The number of students who play Field Hockey (F) and Soccer (S) is = 4

The number of students who play Field Hockey (F) and Basketball (B) is = 2

The number of students who play Soccer (S) and Basketball (B) is = 5

The number of students who play Field Hockey (F), Soccer (S) and Basketball (B) is = 3

The number of students who play nothing = 11

The probability to find the students who play  Field Hockey (F) and Basketball (B) is -

Probability = (number of students who play Field Hockey (F) and Basketball (B)) + (number of students who play Field Hockey (F), Soccer (S) and Basketball (B)) / (Total number of students in survey)

Substitute the values in the equation -

Probability = (2 + 3) / 60

Probability = 5 / 60

Probability = 1 / 12

Probability = 0.083

Therefore, the probability value is obtained as 0.083.

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On solving the equations x + y + z = 6400 and 0.08x + 0.096y + 0.144z = 744, the values for first, second and third investments are $1500, $1700 and $3200 respectively.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

Susan places $6400 in three investments at rates of 8%, 9.6% and 14.4% per annum, respectively.

Let the amount of each investment be x, y and z.

So the mathematical form of the statement is -

x + y + z = 6400

The total income after one year is $744.00.

So the mathematical form of the statement is -

0.08x + 0.096y + 0.144z = 744

The amount placed in the third investment is $1500 more than the amount placed in the second.

So the mathematical form of the statement is -

z = (y + 1500)

Substitute the value of z in first equation -

x + y + (y + 1500) = 6400

x + 2y = 6400 - 1500

x + 2y = 4900

x = (4900 - 2y)

Substitute the value of z in second equation -

0.08x + 0.096y + 0.144(y + 1500) = 744

0.08x + 0.096y + 0.144y + 216 = 744

0.08x + 0.24y = 744 - 216

0.08x + 0.24y = 528

Substitute the value of x in this equation -

0.08(4900 - 2y) + 0.24y = 528

392 - 0.16y + 0.24y = 528

- 0.16y + 0.24y = 528 - 392

0.08y = 136

y = 136/0.08

y = 1700

Therefore, the value of second investment is 1700.

Substitute the value of y in formula for z -

z = (y + 1500)

z = 1700 + 1500

z = 3200

Therefore, the value of third investment is 3200.

Substitute the value of y in formula for x -

x = (4900 - 2y)

x = 4900 - 2(1700)

x = 4900 - 3400

x = 1500

Therefore, the value of first investment is 1500.

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n⁵ + (n+1)⁵ + (n+2)⁵ + (n+3)⁵ + (n+4) is not divisible by 25.

Mathematical Induction is a technique of proving a statement, theorem or formula which is thought to be true, for each and every natural number n.

Given expression,

n⁵ + (n+1)⁵ + (n+2)⁵ + (n+3)⁵ + (n+4)

To prove: above expression is divisible by 25.

Putting n = 1 in the expression

1⁵ + (1+1)⁵ + (1+2)⁵ + (1+3)⁵ + (1+4)

= 1 + 2⁵ + 3⁵ + 4⁵ + 5

= 1 + 32 + 243 + 1024 + 5

= 1305

which is not divisible by 25.

Hence, 25 is not divisor of  n⁵ + (n+1)⁵ + (n+2)⁵ + (n+3)⁵ + (n+4)

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The approximate sales price for a home in 2020 would be 480

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

Year Avg. Sales Price (Thousands)

1990 149

1995 158

2000 207

We can see that there is an upward trend as the year increases

However, the table does not follow a linear or an exponential pattern

From 2000 and 1990, the change in price is 58

Using this as a guide, we have

2020 = 207 + 58 * 2

2020 = 323

The closest to 323 in the option is 480

Hence, the sales price is 480

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The coordinates of the image is A'(3, 4), B'(1, 1), C'(-2, 2) and D'(-3, 4).

A point, line, or geometric figure can be transformed in one of four ways, each of which affects the shape and/or location of the object. Pre-Image refers to the object's initial shape, and Image, after transformation, refers to the object's ultimate shape and location.

Given:

A(-3, -4), B(-1, -1), C(2, -2) and D(3, -4).

In case the quadrilateral ABCD is rotated 180° counterclockwise about the origin, then according to the rule for the rotation of point (x, y) by 180° (x, y) changes into (-x, -y).

Then, the coordinates of the image is

A'(3, 4), B'(1, 1), C'(-2, 2) and D'(-3, 4).

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Answer:

B

Step-by-step explanation:

Should be B.

Derivative of 9X^2 is 18x because the exponent multiplied by the coefficient is 18 and the exponent minus 1 is just 1.

Derivative of 1 is 0 because derivative of any real number is 0

Derivative of y^2 is 2y dy/dx because derivative of y^2 is 2y since the exponent 2 times the coefficient 1 equals 1. And the exponent 2 minus 1 is simply 1.

BUT since it's respect to x, we have to add dy/dx right at the end of the 2y

Answer:

158

Step-by-step explanation:

So the sample size said 15 out of 50. Using the sample size to relate to the total number of students, we get this ratio

[tex] \frac{15}{50} = \frac{x}{526} [/tex]

Let's cross multiply

[tex]50x = (15)(526)[/tex]

[tex]50x = 7890[/tex]

Last, divide by 50 to get X alone

[tex]x = 157.8[/tex]

Which rounds up to 158

The length of the cable is 4810 feet.

We know a right-angled triangle has three sides they are -: Hypotenuse,

Opposite and Adjacent.

We can remember SOH CAH TOA which is,

sin = opposite/hypotenuse, cos = adjecent/hypotenuse and

tan = opposite/adjacent.

If the angle of depression from the taller platform of 12°, Then the angle of elevation is (90 - 12) = 78°.

We know, cos = adjacent/hypotenuse.

cos78° = 1000/hypotenuse.

hypotenuse = 1000/cos78°.

hypotenuse = 4810 feet.

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Answer:

1022

Step-by-step explanation:

cos 12 = 1000/x

The solution:

Exercise 1:  x = -1 and 3.

Exercise 2:  x = 2 and 4.

For variable x : ax² + bx + c = 0, where a≠0 is a standard quadratic equation, which is a second-order polynomial equation in a single variable. It has at least one solution since it is a second-order polynomial equation, which is guaranteed by the algebraic basic theorem.

Given:

The graphs of the two quadratic equation.

So, Exercise 1,

-x² + 2x + 3 = 0.

From the graph,

-1 and 3 are the zeros of the equation.

The solution of the equation is,

x = -1 and 3.

Exercise 2,

x² - 6x + 8 = 0.

From the graph,

2 and 4 are the zeros of the equation.

The solution of the equation is,

x = 2 and 4.

Therefore, the solutions of the both the equation are given above.

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1) The sequence that represents the amount of money left in Isla's bank account at the beginning of each semester is 100,000 - 8,500n.

2) At the end of 10 semesters when Isla graduates, the amount of money left in her account (balance) is $15,000.

The total deposit in Isla's account with interest = $100,000

The withdrawal per semester = $8,500

The number of semesters for Isla to graduate = 10

Let n represent the number of past semesters.

After of money left after each semester can be modeled as 100,000 - 8,500n.

Based on this sequence, after 10 semesters, Isla's account have a balance of $15,000 (100,000 - 8,500(10).

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Answer:

72 mins

Step-by-step explanation:

90 x .8

72

around 125-6

Step-by-step explanation:

The measure of the angles x, y, and z will be 93°, 93°, and 87°, respectively for the rhombus.

The supplementary angles are defined as when pairing of angles addition to 180° then they are called supplementary angles.

As we know that the adjacent angle of the rhombus is supplementary.

As per the given figure, the value of 'x' is given as,

87° + ∠x = 180°

∠x = 180° - 87°

∠x = 93°

Since the opposite angle of the rhombus is congruent to each other.

So ∠y = ∠x

Then ∠y = 93°

And ∠z = 87°

The measure of the angles x, y, and z will be 93°, 93°, and 87°, respectively.

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The total number of pounds is 315 pounds

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

Number of bags = 5

Weight of each bag = 63 pounds

This means that

Total weight = 5 * 63

Evaluate the product

Total weight = 315 pounds

Hence, the total weight is 315 pounds

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For the function,

F(x)  = (x² + 11x +28)/(x² + 8x + 15) limit doesn't exist at x = -5 so it is dis-continuous at that point.

In mathematics, The function is called as continuous at the particular value of x,  if right hand limit, left hand limit and the value of function at x all are equal.

RHL   = LHL  = F(x)

Limit exist and continuous

Given that,

F(x)  = (x² + 11x +28)/(x² + 8x + 15)

To check whether it is continuous at x = -5

First taking LHL,

[tex]\lim_{h \to \ 0}[/tex] [(-5-h)² + 11(-5 -h) +28]/[(-5 -h)² + 8(-5 -h) + 15]

It is approaching at negative infinity

Now, taking RHL

[tex]\lim_{h \to \ 0}[/tex]  [(-5+h)² + 11(-5 +h) +28]/[(-5 +h)² + 8(-5+h) + 15]

Again, It is approaching at negative infinity

Hence, here limit doesn't exist so it is not continuous at x = -5

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Answer:

Step-by-step explanation:

An equivalent method of approximating values of functions is called an approximation using differentials

the answer to this question where is the is c+2x

Answer:

$270

Step-by-step explanation:

h = number of half hours worked

$75 + $45(h-1) = total cost

3 1/2 hrs = 7 half-hours

$75 + $45(7 - 1) = $270

Answer:

89 Bills of 10$ and 1 of 5$

Step-by-step explanation:

Pretty simple division which gets you 89.5 and that decimal is just the one 5$

The price is £12,880.

First, do 14000  x 0.08 = 1,120.

Next do 14,000 - 1,120 = 12,880.

So, the price is £12,880.

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The range of the function is -5 ≤ f(x) ≤ 5.

The correct option is C.

A relation between a collection of inputs and outputs is known as a function. A function is a connection between inputs in which each input is connected to precisely one output. Each function has a range, co-domain, and domain.

A graph of function is given.

The range of the function is all the achievable values.

So, the range of the function is,

-5 ≤ f(x) ≤ 5.

Therefore, the range is -5 ≤ f(x) ≤ 5.

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The sale price of the item is given by the equation A = $ 22.40

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 equation be represented as A

Now , the value of A is

The regular price of the item = $ 28.00

The percentage of discount = 20 %

So , the sale price of the item = regular price of the item  - ( percentage of discount x regular price of the item  )

Substituting the values in the equation , we get

The sale price of the item A = 28.00 - ( 0.20 ) x 28.00

The sale price of the item A = 28.00 ( 1 - 0.20 )

The sale price of the item A = 28.00 ( 0.80 )

So , the sale price of item is 80 percentage of the regular price

Therefore , the sale price of the item = ( 80 / 100 ) x $ 28.00

The sale price of the item A = $ 22.40

Hence , the sale price is $ 22.40

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If a triangle is removed.. from a square of sides 9m. The  area is: 77.25 m².

Area of Square = Side × Side = Side²

Area of Square = 9m × 9m

Area of Square = 81 m²

Area of triangle = 1/2 ×b ×h

Area of triangle = 1/2 × 3m × 2.5m

Area of triangle = 3.75m²

Area of Figure = 81m² - 3.75m²

Area Of Figure = 77.25 m²

Therefore the area is 77.25 m².

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The graph of a quadratic function with vertex (1,-2) has the range and the domain respectively, ( -∞, -2 ) and ( -∞, ∞)

Function is a relation between a set of inputs and a set of outputs which are permissible. In a function, for particular values of x we will get only a single image in y. It is denoted by f(x).

Vertical line test:-

Whenever we want to check whether a given expression is a function or not we can apply a vertical line test, in this test we check for a single image of x , we are getting a single image or more.

If we get more images then it will not be a function.

For example, let us take, y² = 4ax

y = ±√4ax

For single value of x we get two values of y

Hence, it will not be a function.

Given that,

The graph of a function,

whose vertex is (1, -2)

As it can be seen in the graph,

The given quadratic function has defined value for each value of x,

So, the domain of the given function is real numbers,

x  ∈  ( -∞, ∞)

The maximum value of the function is -2

And minimum value of the function is  -∞

The range of the function is   ( -∞, -2 )

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Answer:

x = 75

Step-by-step explanation:

Given the equation:

[tex]\displaystyle{\dfrac{1}{5} = \dfrac{15}{x}}[/tex]

Multiply both sides by 5x to clear off denominators:

[tex]\displaystyle{\dfrac{1}{5}\cdot 5x = \dfrac{15}{x} \cdot 5x}\\\\\displaystyle{1\cdot x = 15\cdot 5}\\\\\displaystyle{x=75}[/tex]

Therefore, x = 75

Transformation of the given function with a feature of the transformed function.

g(x)=-f(x-3) is the vertical asymptote of x=3

j(x)= f(x+3) is x-intercept of (-2,0)

h(x)=3f(x)-3 is domain of (0, ∞)

An asymptote is a line that the graph of a function approaches as either x or y go to positive or negative infinity. There are three types of asymptotes: vertical, horizontal and oblique

Given function is : f(x)= log x

g(x)= -f(x+3)= - log(x+3)

g(x)=-f(x-3) is the vertical asymptote of x=3

j(x)= f(x+3) = log (x+3) then

j(x)= f(x+3) is x-intercept of (-2,0)

Domain: As (0,∞), {x | x>0}

h(x) = 3f(x)-3 is domain of (0, ∞)

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On solving the provided question, we can say that a Formula is a mathematical statement that calculates a value.

In mathematical symbols, a formula is a fact or a rule. Usually, an equal symbol links two or more values. Formulas can be used to determine a quantity's value if you know the value of one. A formula is a collection of mathematical signs and figures that show you how to solve a problem. Examples include formulae for determining the volume of 3D objects and formulas for estimating the perimeter and area of 2D shapes. A mathematical rule or relationship that expresses a variable quantity using letters is called a formula. We refer to these as variables.

A formula is a mathematical statement that calculates a value.

for, example to calculate area of triangle we need a formula.

that formula for area of triangle is = 1/2 * b * h.

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The first-year profits (to the nearest hundred dollars) is, $26,701.18.

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

We have to given that;

In its second year of operation, a local Internet provider's profits were $180,500.

And, This amount was 676% of the company's first-year profits.

Now, Let the amount in the first-year profits = x

So, We can formulate;

⇒ 676% of x = 180,500

⇒ 676/100 × x = 180,500

⇒ x = 180500 × 100 / 676

⇒ x = $26,701.18

Thus, The first-year profits (to the nearest hundred dollars) is, $26,701.18.

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Answer: 62.57 weeks/63 weeks

Step-by-step explanation:

438 divided by 7 = 62.57
Rounded to the nearest week, 62.57 turns to 62.60 which is closer to 63.
The answer is 63 weeks