b) Town A has a rectangular park.
The length of the park is .xm.
The width of the park is 25 m shorter than the length.
The area of the park is 2200 m².
(i) Show that x²-25x-2200 = 0.

In the triangular lattice below, there are an infinite number of equilateral triangles that have at least two of the points as vertices.

This is because the triangular lattice is made up of an infinite number of equilateral triangles of various sizes and orientations. Any two points in the lattice form the vertices of an equilateral triangle and any other point in the lattice can be used as the third vertex.

An equilateral triangle is a triangle in which all three sides are equal in length. It is also equiangular, meaning that all three angles are equal in measure. In the plane, all three vertices of an equilateral triangle can be connected to form a single closed figure. Equilateral triangles can also be used to form periodic patterns, such as the triangular lattice shown in the question.

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Total percent markup in the price of the scooter is 66.67 % ≈ 67%

The Latin phrase "per centum," which meaning "by the hundred," was the source of the English word "percentage." Fractions having a denominator of 100 are called percentages. In other words, it is a relationship where the value of the entire is always assumed to be 100.

A percentage is a ratio or fraction where the full value is always 100. Sam, for instance, would have received 30 out of a possible 100 points if he had received 30% on his arithmetic test. In ratio form, it is expressed as 30:100 and in fraction form as 30/100.

Finding the percentage of a whole in terms of 100 is what percentage calculation refers to. A percentage can be found in one of two ways:

use the unitary approach.

by adjusting the fraction's denominator to 100.

It should be noted that when the denominator is not a factor of 100, the second technique of percentage calculation is not applied. In these situations, we employ the unitary technique.

Cost of Scooter at retailer is $ 75

Cost of Scooter at local Store is $125

Total price markup = 125- 75 = $50

Total percent markup is = 50/75 * 100 = 66.67 % ≈ 67%(approx.)

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The meaning of the y-intercept of the linear function is given as follows:

The tablet starts with 80 percent of battery remaining.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

The graph of the function touches the y-axis at y = 80, hence the intercept b is given as follows:

b = 80.

As the output variable is the battery percentage, it represents the initial battery percentage.

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The meaning of the y-intercept of the linear function is given as follows:

The tablet starts with 80 percent of battery remaining.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

The slope m represents the rate of change of the output variable relative to the input variable.

The intercept b represents the value of y when the graph of the function touches of crosses the y-axis.

The graph of the function touches the y-axis at y = 80, hence the intercept b is given as follows:

b = 80.

As the output variable is the battery percentage, it represents the initial battery percentage.

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The average height of a tree whose age is 25 years old is 364 inches.

A linear model is an equation that describes a relationship between two quantities that show a constant rate of change.

Given that, a table giving a relation between the height and age of trees.

The linear model representing the same is given by,

y = 13.7x + 21.7, where y is the height and x is the age of a tree,

We need to find the average height of a tree whose age is 25 years old,

To find this, put x = 25 in the linear model given,

y = 13.7 × 25 + 21.7

y = 342.5 + 21.7

y = 364.2

y ≈ 364

Hence, the average height of a tree whose age is 25 years old is 364 inches.

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The number of regular employees that got a phone is 39

The number of executive employees that got a phone is 1

Let x represent the number of regular employees

Let y represent the number of executive employees

x + y= 40.......equation 1

50x + 250y= 4000...........equation 2

From equation 1

x= 40-y

Substitute 40-y for x in equation 2

50(40-y) + 250y= 4000

2000- 50y + 250y= 4000

2000 + 2000y= 4000

2000y= 4000-2000

2000y= 2000

y = 1

Substitute 1 for y in equation 1

x + y= 40

x + 1= 40

x= 40-1

x= 39

Hence there are 39 regular employees that got a phone and 1 executive employee that got a phone.

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Rectangle EFGH is assumed, the value of x and m<EIH is 102 degree

Rectangle EFGH is assumed. Each side of a rectangle has an equal length. Each side of a rectangle has an equal length.

FH- =7x-2,

EI-20, and

mZIFG=39°,

Length width equals the area of a rectangle. Therefore, breadth is equal to the product of the area and the length, which is either 20 units or 39 square units.

You need to know a box's height, breadth, and depth in order to determine its volume. These three dimensions may be multiplied to determine the volume.

if < f = 39 degree

so

<G = 39 degree

then

<FIG = 39 + 39 + x = 180 degree

180 - 78 = 102 degree

so <FIG = <EIH

The value of x and m<EIH is 102 degree

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Answer:

An isosceles triangle because it will have two side lengths that are equal.

Step-by-step explanation:

The line in coordinate form passing through the point (−4,−6,6) in the direction is x - y - z + 4 = 0

The term called coordinates in math is called a system that uses one or more numbers, or coordinates and it is uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

Here we have the line in coordinate form passing through the point (−4,−6,6) in the direction.

As we all know that the equation of line through the point (-4, -6, 6) is written as,

=> ( x - (-4)) = (y - (-6))  = (z - 6)

When we simplify this one then we get,

=> x + 4 = y + 6 = x - 6

=> x - y - z + 4 - 6 + 6

=> x - y - z + 4 = 0

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Answer:

V = 8446.9 in³

Step-by-step explanation:

Just use the cylinder formula.

V = πr²h

V = (3.14)(21)²(6.1)

V = 8446.9 in³

Answer: e is the correct option.

Step-by-step explanation:

Option e  If z = xy, then z = 2.

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The solution of the given initial value problem y'' + y = f(t), y(0) = 7, y'(0) = 4 where f(t) = [tex]\left\{\begin{matrix}1, & 0\leq t\leq \frac{\pi}{2}\\ 0, & \frac{\pi}{2}\leq t < \infty\end{matrix}\right.[/tex] is 1 + 6cost + 4sint + u(t - π/2)(cos(t - π/2) - 1).

The initial value problem y'' + y = f(t), y(0) = 7, y'(0) = 6 where

f(t) = [tex]\left\{\begin{matrix}1, & 0\leq t\leq \frac{\pi}{2}\\ 0, & \frac{\pi}{2}\leq t < \infty\end{matrix}\right.[/tex]

Now f(t) = 1[u(t-0) - u(t-π/2)]

f(t) = u(t) - u(t-π/2)

Laplace transformation of f(t) is

L{f(t)} = L{u(t)} - L{u(t-π/2)}

L{f(t)} = 1/S - [tex]e^{\pi S/2}[/tex]/S

Now taking the Laplace transformation on both side to the equation y'' + y = f(t).

L{y''} + L{y} = L{f(t)}

S^2Y(S) - SY(0) - Y'(0) + Y(S) = 1/S - [tex]e^{\pi S/2}[/tex]/S

S^2Y(S) - 7S - 4 +Y(S) = 1/S - [tex]e^{\pi S/2}[/tex]/S

(S^2 + 1)Y(S) - 7S - 4 = 1/S - [tex]e^{\pi S/2}[/tex]/S

(S^2 + 1)Y(S) = 1/S - [tex]e^{\pi S/2}[/tex]/S + 7S + 4

Divide by (S^2 + 1) on both side, we get

[tex]Y(S) = \frac{1}{S(S^2 + 1)} -\frac{e^{\pi S/2}}{S(S^2 + 1)} + \frac{7S}{(S^2 + 1)}+\frac{4}{S^2 + 1}[/tex]

Simplify

[tex]Y(S) = \frac{1}{S}-\frac{S}{S^2 + 1} -\left[\frac{1}{S}-\frac{S}{S^2 + 1}\right]e^{\pi S/2} + \frac{7S}{(S^2 + 1)}+\frac{4}{S^2 + 1}[/tex]

Now Y(t) = [tex]L^{-1}(Y(S))[/tex]

Y(t) = [tex]L^{-1}\left[\frac{1}{S}\right]-L^{-1}\left[\frac{S}{S^2 + 1}\right] -L^{-1}\left[\frac{e^{\pi S/2}}{S}\right]-L^{-1}\left[\frac{Se^{\pi S/2}}{S^2 + 1}\right] + L^{-1}\left[\frac{7S}{(S^2 + 1)}\right]+L^{-1}\left[\frac{4}{S^2 + 1}\right][/tex]

Y(t) = 1 - cost - u(t - π/2) + cos(t - π/2) u(t - π/2) + 7cost + 4sint

Y(t) = 1 + 6cost + 4sint + u(t - π/2)(cos(t - π/2) - 1)

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The complete question is:

Find the solution of the given initial value problem y'' + y = f(t), y(0) = 7, y'(0) = 4 where f(t) = [tex]\left\{\begin{matrix}1, & 0\leq t\leq \frac{\pi}{2}\\ 0, & \frac{\pi}{2}\leq t < \infty\end{matrix}\right.[/tex]

Answer: y=-2x

Step-by-step explanation:

take formula (y-y1)=m(x-x1), m is slope

plug in x and y values and slope, (y-7)=-2(x-10)

distribute, y-7=-2x+20

y=-2x+27

If there were 132 crayons in John's backpack, the number of blue crayons is equal to blue crayons.

In Mathematics, a proportion simply refers to an equation which is typically used to represent the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.

By applying direct proportion to the given information, we have the following mathematical expression:

3 crayons = 8 blue crayons

132 crayons = X blue crayons

By cross-multiplying, we have the following:

3x = 132 × 8

3x = 1,056

x = 1,056/3

x = 352 blue crayons.

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Answer:

x= 1, y=3

Step-by-step explanation:

x + (2x+1)= 4

3x+1=4

3x=4-1

x=1

Take x as 1

1 + y =4

y=4-1

y=3

Answer: yes

2x^2yx(x^2+2y)

Step-by-step explanation:

2x^4y+4x^2y^2

2x^2yx(x^2+2y)

The sign of the derivative of a function at a certain point indicates the nature of the function's change at that point.

If the derivative is positive at a point, the function is increasing at that point. If the derivative is negative at a point, the function is decreasing at that point. If the derivative is undefined at a point, there may be a sharp turn or a discontinuity at that point.

Here's a more specific method to determine the sign of the derivative at a point:

It's also possible to determine the sign of the derivative by analyzing the graph of the function, or by using calculus theorems such as the First Derivative Test or the Second Derivative Test.

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The approximate number of time does an average human heart beat in a lifetime is 3 x 10⁹

The term called average in math is defined as the mean value which is equal to the ratio of the sum of the number of a given set of values to the total number of values present in the set.

Here we need to find the number of time does an average human heart beat in a lifetime.

As we all know that a normal heart rate when you're not being active is generally between 60 – 100 beats per minute.

And this is called your resting heart rate.

Based on this we have calculated as average of 80 beats per minute, your heart beats about 4,800 times per hour. That's a whopping 115,200 times per day.

Then during an average lifetime, then the human heart will beat more than 2.5 billion times.

When we round off the value then we get approximately 3 billon.

In a scientific notation it can be written as,

=> 3 x 10⁹

Therefore option (c) is correct.

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Step-by-step explanation:

g(x) is f(x) shifted up by 2 units.

that is simply expressed as

g(x) = f(x) + 2

h(x) if f(x) shifted left by 1 unit.

that is trickier.

h(x) = f(x + 1)

why ?

every result that f(x) had at x+1, is now happening for h(x) at x ("earlier", shifted to the left). so, the function result for f(x+1) is the function result for h(x).

and therefore,

h(x) = f(x + 1)

FYI

a shift to the right by 1 unit would be

i(x) = f(x - 1)

for the same reasons as for the shift to the left, just in the other direction.

The probability that the two selected chips are of different colors is 15/32

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

Blue = 5

Yellow = 3

This means that

Total = 5 + 3

Total = 8

The probability of different color is then calculated as

P(Different) = Yellow and Blue or Blue and Yellow

So, we have

P = 3/8 * 5/8 * 2

Evaluate

P = 30/64

Simplify

P = 15/32

Hence, the probability is 15/32

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Answer:

300

Step-by-step explanation:

1.5*200 = 300

Cool.

That m is odd, then the integers between 1 and m − 1 equals 2−1 mod m it must be k=1 and n = 1+m2.

Given an integer m, two integers a and b are congruent modulo m if

m| (a − b).

We write a ≡ b (mod m).

Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers. The inverse additives of the equivalent positive numbers are the negative numbers.

I will also sometimes say equivalent modulo m. Notation note: we are using that "mod" symbol in two different ways.

Let n be the inverse of 2( mod m)

Where 1<n<m−1

2n≡1 ( mod m)

⇒ 2n=1+mk

Since m is odd then m=2l+1

⇒ 2n=1+(2l+1)k

Then solve for n(mod m) as that would put it within 1<n<m−1.

As an aside, we only care for 1≤n≤m−1, we don't care about strict inequalities. Indeed, if m=3 then you'd have a problem as there aren't even any integers strictly between 1 and 3−1

Supposing that m = [tex]2l+1[/tex] then [tex]2*(l+1)[/tex] = [tex]2l + 2[/tex] = m+1

We want integer n=1+mk2 between 1 and m−1.

Since m is odd, we need k to be odd, in order for 1+mk to be divisible by 2 so n is an integer.

If k≥3 then 1+mk2>m−1.

If k≤−1 then 1+mk2<1.

So it must be k=1 and n = 1+m2.

Therefore,

That m is odd, then the integers between 1 and m − 1 equals 2−1 mod m it must be k=1 and n = 1+m2.

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Therefore , the solution of the given problem of fraction  comes out to be

height h = 3 meters.

Part of a whole is a fraction. The quantity is written as a quotient in mathematics, where the numerator and denominator are divided. Each is an integer in a simple fraction.

Here,

Given :

=> 48π m².

=> 5π meter and radius = 8/2 = 4 meter

=>π(4)(4)h =48π

=> 16πh = 48π

=> h = 3 meters

Whether  it is in the numerator or denominator, a complex fraction contains a fraction. The numerator and denominator of a correct fraction are opposite each other.

Therefore , the solution of the given problem of fraction  comes out to be

height h = 3 meters.

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Answer: 0.068

Step-by-step explanation:

i used a calculator so I think it's right

B) The Rational Root Theorem may be used to identify the potential rational roots of the equation in Part A.

The divisors of 9 with the opposite sign are possibly rational roots.

Then, for each rational root, we can see if it is a factor of the polynomial.

A) The remaining of f(x) divided by (x - a) is R, while the remainder of f(x) divided by (x - a) is 2R.

As a result, we may write the following polynomials:

f(x) = (x - a) (x - a)

Q(x) + R f(x) = (x - a)

Q(x) + 2R

Where Q(x) represents the quotient and R represents the remainder.

By combining the two polynomials, we obtain:

(x - a)

Q(x) + R = (x - a) (x - a)

Q(x) + 2R

When we subtract (x - a)Q(x) from both sides, we get:

R = 2R

R = 0 when we solve for R

As a result, the remaining when f(x) is divided by (x - a) is zero.

That is, (x - a) is a factor of f(x), and f(x) is represented as:

f(x) = (x - a)(x^2 + bx + c)

Comparing the coefficients, we get:

a = -a

b = 2a

c = -9

So we have the following equation:

x^3 - 2x^2 - 18x - 9 = 0

B) If it is a factor, we may use synthetic division to divide the polynomial by the factor, resulting in a new polynomial, which we can continue until we have a polynomial with no rational roots. We can utilize numerical methods such as the bisection method or the Newton-Raphson method to thoroughly solve the problem.

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Answer:

20-15-60 is the answer to the question

Answer:

Ben's savings (S) can be related to the number of hours he babysits (h) by the equation:

S = r * h

where r is the rate of pay per hour that Ben receives for babysitting. This equation shows that Ben's savings increase linearly with the number of hours he babysits, as long as the rate of pay remains constant.

The required average temperature of the last summer is given as 22.10 °C.

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

Here,
Let the temperature of last summer's average be x insert,

According to the question,
21 = x - 5%x
x = 21 / 0.95
x = 22.10 °C

Thus, the required average temperature of the last summer is given as 22.10 °C.

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The annual interest rate of the deposit is 1.5% per annum.

Borrowers must pay lenders simple interest as a fee in exchange for a loan. Compound interest is excluded from the calculation and just the original principal is used. Simple interest applies to all loans, not just specific ones. Additionally, it is the kind of interest that banks give their customers on their savings accounts.

Given amount deposit = $500

interest earned in 8 years = $60

time = 8 years

simple interest is given by,

SI = (Prt)/100

P = principle = $500

r = rate per annum

t = time = 8 years

60 = (500*8*r)/100

60*100 = 4000r

r = 6000/4000

r = 1.5% per annum

Hence rate is 1.5% per annum.

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By the compare to y =mx + b, the value of slope (m) is 1/10 and value of y - intercept is 0.

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

We have to given that;

The expression is,

⇒ v = 1/10u

Now, We can write as;

⇒ v = 1/10u

⇒ v = 1/10u + 0

By compare with y = mx + b, we get;

⇒ slope (m) = 1/10

⇒ y - intercept (b) = 0

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