Help with math problems

The term Exponential decay in mathematics explains as a process of a constant percentage rate reduction in an amount over time.

In elementary mathematics, the graph designates a plot or a function graph. A graph is a set of points and the connections between some subset of those points (which may be empty).

Asymptote, An object that serves as the limit of another line or curve in mathematics.

y-intercept is (0, −2) means x = 0 and y = -2

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The absolute value function f(x) = |4x + 1| is defined as follows:

For any x in the domain of f(x),

f(x) = 4x + 1 if 4x + 1 ≥ 0, i.e., if x ≥ -1/4

f(x) = -(4x + 1) if 4x + 1 < 0, i.e., if x < -1/4

Therefore, the graph of f(x) consists of a line with slope 4 and y-intercept 1 for x ≥ -1/4, and a line with slope -4 and y-intercept -1 for x < -1/4. At x = -1/4, the graph has a sharp turn, which is called a corner point.

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Answer:

1)  You should add 49 to both sides to complete the square.

    [tex]\textsf{Solutions:}\;\;\;x=3,\;\;x=-17[/tex]

2)  The discriminant of the equation is 261.

    There are 2 solutions.

    [tex]\textsf{Solutions:}\;\;\;n=\dfrac{1 +3\sqrt{29}}{10},\;\;n=\dfrac{1 -3\sqrt{29}}{10}[/tex]

Step-by-step explanation:

Given quadratic equation:

[tex]x^2+14x-51=0[/tex]

To solve the given quadratic equation by completing the square, first move the constant to the right side of the equation:

[tex]\implies x^2+14x=51[/tex]

Add the square of half the coefficient of the term in x to both sides:

[tex]\implies x^2+14x+\left(\dfrac{14}{2}\right)^2=51+\left(\dfrac{14}{2}\right)^2[/tex]

Simplify:

[tex]\implies x^2+14x+\left(7\right)^2=51+\left(7\right)^2[/tex]

[tex]\implies x^2+14x+49=51+49[/tex]

[tex]\implies x^2+14x+49=100[/tex]

Therefore, we added 49 to both sides of the equation to complete the square.

To solve, factor the perfect square trinomial on the left side of the equation:

[tex]\implies (x+7)^2=100[/tex]

Square root both sides:

[tex]\implies \sqrt{(x+7)^2}=\sqrt{100}[/tex]

[tex]\implies x+7=\pm10[/tex]

Subtract 7 from both sides of the equation:

[tex]\implies x=-7\pm10[/tex]

Therefore, the solutions are:

[tex]x=3,\;\;x=-17[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Quadratic Formula}\\\\$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$\\\\when $ax^2+bx+c=0$ \\\end{minipage}}[/tex]

Given quadratic equation:

[tex]5n^2-n=13[/tex]

To solve using the quadratic formula, subtract 13 from both sides so that the equation is in the form ax² + bx + c = 0:

[tex]\implies 5n^2-n-13=0[/tex]

Therefore:

The discriminant of the equation is b² - 4ac:

[tex]\begin{aligned}\implies b^2-4ac&=(-1)^2-4(5)(-13)\\&=1+260\\&=261\end{aligned}[/tex]

Therefore, the discriminant is 261.

As the discriminant is positive, there are 2 solutions.

To find the solutions, substitute the values of a, b and c into the quadratic formula and solve for n:

[tex]\implies n=\dfrac{-(-1) \pm \sqrt{(-1)^2-4(5)(-13)}}{2(5)}[/tex]

[tex]\implies n=\dfrac{1 \pm \sqrt{1+260}}{10}[/tex]

[tex]\implies n=\dfrac{1 \pm \sqrt{261}}{10}[/tex]

[tex]\implies n=\dfrac{1 \pm \sqrt{9\cdot29}}{10}[/tex]

[tex]\implies n=\dfrac{1 \pm3\sqrt{29}}{10}[/tex]

Therefore, the solutions are:

[tex]n=\dfrac{1 +3\sqrt{29}}{10},\;\;n=\dfrac{1 -3\sqrt{29}}{10}[/tex]

To find f(g(x)), we substitute g(x) in place of x in the function f(x).

f(g(x)) = f(5x/(1-x)) = (5x/(1-x)) / (5 + (5x/(1-x))) = 5x / (1-x+5-5x) = 5x / 6 - 4x/6 = x/6

To find g(f(x)), we substitute f(x) in place of x in the function g(x).

g(f(x)) = g(x/(5+x)) = (5(x/(5+x))) / (1 - (x/(5+x))) = 5x / (5+x-x) = 5x/5 = x

Therefore, f(g(x)) = x/6 and g(f(x)) = x.

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We have shown that line segment BDE is the perpendicular bisector of line segment AC.

A perpendicular bisector is a line that intersects a given line segment at its midpoint and forms a right angle with that line segment. In other words, if you draw a line that cuts a line segment in half at a 90-degree angle, that line is called the perpendicular bisector of the line segment. The perpendicular bisector of a line segment is important in geometry because it is one of the ways to construct a line that is equidistant from the endpoints of the line segment. This property makes it useful in various applications, such as in the construction of regular polygons, and in determining the circumcenter of a triangle.

to show that line segment BDE is the perpendicular bisector of line segment AC, which means we need to show that BDE:

is perpendicular to AC, and

bisects AC, i.e. intersects it at its midpoint.

First, we note that since angle ABE is congruent to angle CBE, and angle ADE is congruent to angle CDE, we have:

angle ABE + angle ADE = angle CBE + angle CDE

This implies that angles BAE and CDE are congruent (since they are corresponding angles formed by parallel lines AB and CD intersecting transversal AE), and angles BDE and CAE are congruent (since they are vertical angles).

Therefore, we have the following:

angle AED = angle CEB (vertically opposite angles)

angle BDE = angle CAE (vertically opposite angles)

angle ABE = angle CBE (given)

angle ADE = angle CDE (given)

angle BAE = angle CDE (sum of angles)

Now, we proceed to show that BDE is perpendicular to AC. Let M be the midpoint of AC, and let BD and BE intersect AC at P and Q, respectively.

Since M is the midpoint of AC, we have:

AP = PC

AQ = QC

Also, since BAE is congruent to CDE, we have

BA = CD

Therefore, we have:

AP + PB = AC = CQ + QB

AP + PB = AQ + QB

PB = AQ

So, P and Q are equidistant from M, which means that line segment BDE passes through the midpoint of AC. Therefore, we have shown that BDE bisects AC.

To show that BDE is perpendicular to AC, we note that angles BDE and CAE are congruent, and angles AED and CEB are congruent. Therefore, triangles AED and CEB are similar by angle-angle similarity. Since corresponding sides of similar triangles are proportional, we have:

BE/EA = CB/EB

This implies that:

BE² = EA * CB

Similarly, we can show that:

BD² = DA * CB

Adding these two equations, we get:

BE² + BD² = (EA + DA) * CB = AC * CB

Therefore, we have:

BE² + BD² = AC * CB

Since we have already shown that BDE bisects AC, we know that CB = 2 * BM. Substituting this into the equation above, we get:

BE²+ BD² = 2 * AC * BM

Therefore, line segment BDE is perpendicular to AC, as desired.

Therefore, we have shown that line segment BDE is the perpendicular bisector of line segment AC.

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Answer:

Step-by-step explanation:

C IS CORRECT

Answer:

A little more than happy for me and then the other

Answer: 1/9, or 3^-2

Step-by-step explanation: -3^-2 is equivalent to 1/-3^2, which is 1/9

Answer:

-11

Step-by-step explanation:

[tex]1. \: - 9 - 2 \\ 2. \: - 11[/tex]

The value of x for the given equation is - 3.11.

The two most well-known groups of equations in algebra are the linear equations and the polynomial equations. P(x) = 0 can be used to represent polynomial equations with a single variable. P is a polynomial, and axe + b = 0 is the standard form for linear equations. Here, the parameters a and b. To solve these equations, we can use geometric or computational techniques from linear algebra or mathematical analysis. There are several sorts of equations as well, including: linear algebra, quadratic formulas, cube-based equations, etc.

Given that, the equation is x = 8/9 - 4.

Take the LCM.

x = 8/9 - 36/9

x = (8 - 36) / 9

x = - 3.11

Hence, the value of x for the given equation is - 3.11.

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Answer:

The equation given is in point-slope form, where the point (4,8) is on the line and the slope is undefined.

To solve for y, we can first isolate the term with y on one side of the equation:

y - 8 = (1/0)(x - 4)

Since 1/0 is undefined, this equation is not solvable for y using algebraic methods.

The graph of this equation would be a vertical line passing through the point (4,8), since the slope is undefined.

Step-by-step explanation:

The probability that a person from the East Coast has no preference can be found to be 17 %.

The probability that a person from the East Coast does not have a preference between regular soda and diet soda can be found out by the formula :

= Number of people from East coast without a preference / Total people surveyed in the East Coast x 100 %

Number of people from east coat without a preference = 17 people

Total people surveyed in the East Coast = 100

The probability then is :

= 17 / 100 x 100 %

= 17 %

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The question is :

What is probability that a person from the East Coast has no preference?

A marketing director for a beverage company conducted a study to investigate peoples soda preferences in two regions of the country. The director selected a random sample of 100 people from the east coast and a random sample of 100 people from the west coast to survey. The responses are summarised in the following table.

Answer:

2.50

There is a total of 7 items that cost 17.50

17.50 divided by 7 is 2.50

Answer:

A pretzel and a drink cost the same: $2.50

Step-by-step explanation:

[tex]3p+4d=17.50[/tex]

[tex]p=d[/tex]

Then:

[tex]3(d)+4d=17.50[/tex]

[tex]3d+4d=17.50\\7d=17.50[/tex]

[tex]d=17.50/7[/tex]

[tex]d=2.50[/tex]

[tex]p=d=2.50[/tex]

Hope this helps.

634 is 79.25% of 800.

let x be the percent

(x/100)*800= 634

or, x*8=634

or, x= 634/8 = 79.25
Therefore, 634 is 79.25% of 800.

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Answer: x=5±sqrt2

Step-by-step explanation:

(x-5)^2=2 take the principal and negative root of both sides
x-5=±sqrt2 add 5 to both sides
x=5±sqrt2

Answer: x = 5 [tex]\pm\sqrt{2}[/tex]

Step-by-step explanation:

      To solve, we will isolate the variable x.

   Given:

(x - 5)² = 2

   Square root both sides of the equation:

x - 5 = [tex]\pm\sqrt{2}[/tex]

   Add 5 to both sides of the equation:

x = 5 [tex]\pm\sqrt{2}[/tex]

Answer:

The ball with diameter 8 ( radius 4) is
8 times bigger
than the ball with diameter 4 (radius  2)

Step-by-step explanation:

We will use volume as an indication of bigger and smaller

For a spherical ball, volume

[tex]V = \dfrac{4}{3}\pi r^3\\[/tex]

where r is the radius

Since 4/3 and π are constants we can rewrite this equation as
V = k r³

where k = 4/3 x π

We see that the greater the radius, the greater the volume

First note that diameter = r/2
So ball with diameter 8 has radius 8/2 = 4

and ball with diameter 4 has radius 4/2 = 2

Volume of ball with radius 4 = V₄ = k x 4³ = 64k

Volume of ball with radius 2 = V₂ = k x 2³ = 8k

V₄/V₂ =64k/8k = 8

So the ball with radius 4 (diameter 8) is 8 times bigger than the ball with radius 2(diameter 4)

---------------------------------------------------------------------------------------------------------

Another way to understand this is that the diameter of the larger ball is 2 x the diameter of the smaller ball.

Hence the radius of the larger ball is also 2 x the radius of the smaller ball

Therefore larger to smaller volume = 2³ =8

Answer:

Step-by-step explanation:

To find the lower quartile, we need to first order the data set from smallest to largest:

{1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5}

The lower quartile is the median of the lower half of the data set. Since there are an even number of data points, we take the average of the middle two values:

Lower quartile = (2+2)/2 = 2

Therefore, the lower quartile for this data set is C. 2.

Answer:

rewrite 0,26 as a proper fraction (in the format of a over b) show all steps

The graphs of the variable g(x) is represented by the equation g(x) Equals [tex]f(x+3)-2[/tex].

Your example is an equation since an equations is any statement with an equals sign. Due of mathematicians' fondness of equal signs, equations are commonly used in mathematical expressions. A equation is a collection of guidelines for producing a certain outcome.

A formula that describes the relationship between two statements on either side of a sign is known as a simple equation. Usually, it has single parameter as well as an equal sign. akin to this 2. 2x - 4 = 2.

The equation of the function f(x) is given as:

[tex]f(x)=x^{3}[/tex]

The function g(x) is shifted down by 2 units and shifted left  by 3 units, to get the function g(x)

This means that:

[tex]g(x)=f(x+3)-2[/tex]

So, we have:

[tex]g(x)=f(x+3)-2[/tex]

Hence, the equation that represents the graph of the function g(x) is g(x)

[tex]=f(x+3)-2[/tex]

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The complete question is:

If f(x) = [tex]x^{3}[/tex], what is the equation of the graphed function?

A. y = f(x-3) - 2

B. y = f(x+3) - 2

C. y = f(x+2) - 3

D. y = f(x-2) + 3

The two numbers such that the distance between (2, -1) and (t, 3) equals 7​ are

The distance between two points (x, y) and (x', y') is given by

d = √[(x' - x)² + (y' - y)²]

Since we need to find two numbers for such that the distance between (2, -1) and (t, 3) equals 7., then

So, substituting the values of the variables into the equation, we have that

d = √[(x' - x)² + (y' - y)²]

7 = √[(t - 2)² + (3 - (-1))²]

√[(t - 2)² + (3 + 1)²] = 7

√[(t - 2)² + 4²] = 7

√[(t - 2)² + 16] = 7

Squaring both sides, we have that

[(t - 2)² + 16] = 7²

(t - 2)² + 16 = 49

(t - 2)² = 49 - 16

(t - 2)² = 33

Taking square root of both sides, we have that

t - 2 = ±√33

t = 2 ± √33

So, t = 2 + √33 and t = 2 - √33

So, the two number are

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The dimensions of the rectangle are length 2 and width 8. The solution has been obtained by using the concept of derivatives.

What is a derivative?

The derivative of a quantity in mathematics refers to the variable's instantaneous rate of change with respect to another.

We are given y = 12 - x² which is an even function. This means that the rectangle form will also be even at the origin.

We know that the

Area of rectangle = length × width

So, here length = 2x and width = y

Using the formula, we get

⇒Area (A) = 2x (12 - x²)

⇒A = 24x - 2x³

Now, on differentiating the area, we get the derivative as

A’ = 24 - 6x²

We know that the area will be largest when A’ = 0

On substituting, we get

⇒ 24 - 6x² = 0

⇒ 6x² = 24

⇒ x² = 4

⇒ x = 2

Now, on substituting this value in y = 12 - x², we get

⇒ y = 12 - 4

⇒ y = 8

Hence, the dimensions of the rectangle are length 2 and width 8.

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The height of the frame 20 cm from the center of the base is 32 cm.

Since the window frame is shaped like half of an ellipse, we know that the vertical height of the frame changes as we move horizontally along the base. The height of the frame at any point along the base can be calculated using the equation for the vertical axis of an ellipse:

h(x) = b * sqrt(1 - (x^2 / a^2))

where h(x) is the height of the ellipse at a given horizontal position x, a is the horizontal radius (half of the width), and b is the vertical radius (half of the height).

In this case, we know that the height of the frame at the center of the base (x = 0) is 40 cm, and we are interested in finding the height of the frame 20 cm from the center of the base (x = 20 cm). We also know that the width of the frame (2a) is 160 cm.

Using these values, we can solve for the vertical radius b:

b = h(x=0) = 40 cm

Then we can use the equation above to find the height of the frame 20 cm from the center of the base:

h(x=20) = b * sqrt(1 - (x^2 / a^2)) = 40 * sqrt(1 - (20^2 / 80^2)) = 32 cm

Therefore, the height of the frame 20 cm from the center of the base is 32 cm.

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Answer:

0.84

Step-by-step explanation:

The probability of a value being greater than 104 in a normal distribution with a mean of 98 and a standard deviation of 6 can be calculated using the z-score formula. The z-score for 104 is (104-98)/6 = 1. The probability of a value being greater than 104 is equal to 1 minus the cumulative probability of the Z-score, which is 0.84.

Answer:

55.1372 m

Step-by-step explanation:

The composite figure consists of two regular figures

a rectangle of length 16 m and width 9 m

and

a semicircle of diameter d = 9 m

The total length of the outside of the rectangle = 16 + 16 + 9 = 41 m

Circumference of a circle = πd = 9π

Length of the semicircle portion is 1/2 this = 0.5 x 9π = 14.1372 m

Total perimeter =  41 m +  14.1372 m = 55.1372 m

The solution to the division problem shown is 2x² + x + 2 + 1/(4x+3)

Polynomial expression are expression that has a leading degree of 3. Given the expression below:

(8x³+10x² +15x+10)/(4x+3)

We can use polynomial long division to divide (8x³+10x²+15x+10) by (4x+3):

      2x² + x + 2    

------------------------

4x+3| 8x³ + 10x² + 15x + 10

8x³ + 6x²

-----------

4x² + 15x

4x² + 3x

---------

12x + 10

12x + 9

--------

1

Therefore, the quotient is 2x² + x + 2 and the remainder is 1. So:

(8x³+10x²+15x+10)/(4x+3) = 2x² + x + 2 + 1/(4x+3)

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Answer:

Step-by-step explanation:

(a) Let P(n) denote the population of the town at the start of year n.

The percentage change in population due to births and deaths is given by:

(0.09 - 0.005) = 0.085 or 8.5% per annum.

So, the population at the start of year (n+1) will be:

P(n+1) = P(n) + 0.085P(n)

In addition, a constant number of 250 new residents move into the town each year. Therefore, the population at the start of year (n+1) will be:

P(n+1) = P(n) + 0.085P(n) + 250

Combining these two equations, we obtain the desired recurrence relation:

P(n+1) = 1.085P(n) + 250

(b) To find a formula for the population of the town at the start of the nth year, we can use the recurrence relation and iterate it n-1 times. We have:

P(1) = 3000 (given)

P(2) = 1.085(3000) + 250 = 3385

P(3) = 1.085(3385) + 250 = 3802.25

P(4) = 1.085(3802.25) + 250 = 4262.36

and so on.

Thus, the population of the town at the start of the nth year is given by the formula:

P(n) = (3000 + 250/0.085) * 1.085^(n-1) - 250/0.085

A $1,000 computer in Williamson County, Tennessee would cost $1,092.50 after including the sales tax.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

The sales tax rate in Tennessee is 7%, but the rate in Williamson County is 2.25% higher, making it 9.25%.

To calculate the cost of a $1,000 computer in Williamson County, you would need to add the sales tax to the base price:

Sales Tax = Base Price x Sales Tax Rate

Sales Tax = $1,000 x 0.0925

Sales Tax = $92.50

Therefore, the total cost of the computer in Williamson County would be:

Total Cost = Base Price + Sales Tax

Total Cost = $1,000 + $92.50

Total Cost = $1,092.50

Therefore, a $1,000 computer in Williamson County, Tennessee would cost $1,092.50 after including the sales tax.

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Answer:

The two numbers that have a difference of 4 and a product of 140 are:

Step-by-step explanation:

A factor pair is a set of two integers that when multiplied together, give a particular product.

List the factor pairs of 140:

Calculate the difference between the factor pairs by subtracting the smaller number from the larger number:

Therefore, the two numbers that have a difference of 4 and a product of 140 are:

Answer:

$7.15

Step-by-step explanation:

Let $X be a cost of fuel for travel 65 km

and it is given that $4.95 is cost of fuel to travel 45 km

[tex]\frac{X}{4.95} = \frac{65}{45}[/tex]

Multiply both side by 4.95 we get ,

[tex]X = \frac{65}{45}[/tex]×[tex]4.95[/tex]

Multiply botth numerator and denominator in RHS by 100 to remove decimal in 4.95

X = 65/45 × 495/100

X = 65/100 × 495/45

X = 65/100 × 11                        ........(495/45 = 11)

X = 0.65 × 11

X = 7.15

To travel 65 km the cost of fuel is $7.15