SOLVE THE FOLLOWING INEQUALITY: x^3-5x^2+10 > 0

By the Principle of Mathematical Induction, 2ⁿ≥2n is true for all positive integers.

The given inequality is 2ⁿ≥2n.

Let n = 1

2^1 ≥ 2^1

2 ≥ 2 which is true

Inductive Step: Assume 2ⁿ≥2n is true for some arbitrary positive integer k.

We need to prove that 2^(k+1) ≥ 2^(k+1)

2^(k+1) ≥ 2*2^k (by the inductive hypothesis)

2^(k+1) ≥ 2*2^(k+1)

2^(k+1) ≥ 2^(k+1) which is true

Therefore, by the Principle of Mathematical Induction, 2ⁿ≥2n is true for all positive integers.

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Answer:

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See attached graph to help with answer choices.

Given function:

Answer choices:

In the quadratic formula, the number for "a" is filled in with the coefficient of x².

In Mathematics and Geometry, a quadratic equation can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.

In Mathematics, the standard form of a quadratic equation is represented by the following equation;

ax² + bx + c = 0

Mathematically, the quadratic formula is represented by this mathematical equation:

[tex]x = \frac{-b\; \pm \;\sqrt{b^2 - 4ac}}{2a}[/tex]

For instance, given the quadratic equation 2x² - 3x - 1 = 0, we have:

a = 2

b = -3

c = -1

[tex]x = \frac{-(-3)\; \pm \;\sqrt{(-3)^2 - 4(2)(-1)}}{2(2)}\\\\x = \frac{3\; \pm \;\sqrt{9 + 8}}{4}\\\\x = \frac{3\; \pm \;\sqrt{17}}{4}[/tex]

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The height of the triangle is √(115) - 1 inches and the width is (1 + √(115)) / 6 inches.

Let's begin by assigning variables to the height and width of the triangle. We'll use h for height and w for width.

From the problem statement, we know that the area of the triangle is 184 square inches:

Area = (1/2) * base * height

where the base is equal to the width. We can rearrange this formula to solve for the height:

height = 2 * Area / base

Since the area is given as 184 square inches and the base is equal to the width, we can write:

h = 2 * 184 / w

We also know that the height is two less than six times the width. Writing this as an equation, we have:

h = 6w - 2

Now we can substitute the expression for h from the second equation into the first equation:

2 * 184 / w = 6w - 2

Multiplying both sides by w gives:

2 * 184 = w * (6w - 2)

Expanding the right side gives:

2 * 184 = 6w² - 2w

Simplifying further gives:

6w² - 2w - 368 = 0

This is a quadratic equation that we can solve using the quadratic formula:

w = (-b ± √(b² - 4ac)) / 2a

where a = 6, b = -2, and c = -368. Plugging in these values gives:

w = (2 ± √(2² - 4 * 6 * (-368))) / 2(6)

Simplifying further gives:

w = (1 ± √(115)) / 6

Taking the positive value gives:

w = (1 + √(115)) / 6

Plugging this value back into either equation for h gives:

h = 6w - 2 = 6((1 + √(115)) / 6) - 2 = 1 + √(115) - 2 = √(115) - 1

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Answer:

93 + 7 = 100

Step-by-step explanation:

I would assume positive because 100 is above the x-axis not below it.

Answer:

  a)  |x -8| = 6

  b)  |x -15| = 0

Step-by-step explanation:

You want the values of 'b' and 'c' for the cases where the solutions to the equation |x -b| = c are ...

The solutions to |x -b| = c are the solutions to ...

  x -b = c   ⇒   x = b +c

  x -b = -c   ⇒   x = b -c

Given the two solutions P and Q, the values of 'b' and 'c' can be found from ...

  P = b +c

  Q = b -c

Adding these two equations gives ...

  P +Q = 2b   ⇒   b = (P +Q)/2

Subtracting the second equation from the first gives ...

  P -Q = 2c   ⇒   c = (P -Q)/2

  b = (2 +14)/2 = 8

  c = (14 -2)/2 = 6

The equation is ...

  |x -8| = 6

  b = (15 +15)/2 = 15

  c = (15 -15)/2 = 0

The equation is ...

  |x -15| = 0

Expressing the expansion of the logarithm in terms of log y, we can write: log(8x^2) = log(8) + 2*log(y), where y = x

Using the logarithmic identity that states log(a^b) = b*log(a), we can expand log(8x^2) as:

log(8x^2) = log(8) + log(x^2)

Note that a  logarithmic identity is a mathematical relationship or equation that involves logarithms.

We can further simplify log(x^2) as:

log(x^2) = 2*log(x)

Therefore, we can write:

log(8x^2) = log(8) + 2*log(x)

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Answer:

Family: exponential functions

Step-by-step explanation:

The given function is:

[tex]y = 3*2^x - 6[/tex]

It is in the form of exponential function:

[tex]y = a*b^x + c[/tex]

where "a", "b", and "c" are constants.

The parent function of an exponential function is:

[tex]y = b^x[/tex]

where "b" is the base of the exponential function. In this case, the parent function would be:

The given function is a member of the family of exponential functions with base 2 and with a vertical shift of -6 units and a vertical stretch by a factor of 3.

Answer:

Exponential function family with base 2.

[tex]\textsf{Parent function:} \quad y=2^x[/tex]

Step-by-step explanation:

The given equation is in the form of an exponential function with a vertical stretch and vertical shift.

The standard form of this type of exponential function is:

[tex]y=ab^x+c[/tex]

where:

If |a| > 1 then it is a vertical stretch, and if 0 < |a| < 1 it is a vertical "compression".

The parent function of this type of exponential function is y = bˣ, where b is the base.

Given function:

[tex]y=3 \cdot 2^x-6[/tex]

Therefore, the parent function of the given function is:

[tex]\boxed{f(x)=2^x}[/tex]

The given function is a vertical stretch by a factor of 3, followed by a vertical shift downwards by 6 units, from the parent exponential function with base 2.

Hence, the family of functions that this equation belongs to is the exponential function family with base 2, and the parent function is [tex]y = 2^x[/tex].

A 3:5 scaled version means that the dimensions of the new version are proportional to the original dimensions in a ratio of 3:5.

A scaled version is a version of a shape where every size in the original shape is multiplied by the same number.

In this situation, a 3:5 scaled version means that the dimensions (or sizes) of the new version are proportional to the original dimensions in a ratio of 3:5.

For example, if we have a circle with radius 20 cm, we can create a 3:5 scaled version by multiplying the radius by the ratio 3/5. That is:

scaled version radius = 3/5 * 20 = 12 cm

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The equation which can be used to find the amount the admission tickets cost would be 4x + 25 = 175. The cost of the admission tickets would therefore be $ 37.50.

The total spent by Ms. Boi was $ 175.00 so this is what the entire equation would be equal to. The cost of admission tickets will be denoted as x because they are the unknown figure.

There are 4 tickets purchased so the equation would be:

4 x Cost of tickets - Parking cost = Total spent

4 x - 25 = 175

The cost of each admission ticket is:

4 x - 25 = 175

x = 150 / 4

x = $ 37.50

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Answer:is b

A line segment has both of its end-points fixed and so it has a definite length.

Step-by-step explanation:

pl z give me brianlieyst and have a good day

:)

Answer:

C

Step-by-step explanation:

The equation is:

y = [tex]\frac{1}{4}[/tex]x - 8

If you plug in the x and y value from the table, you find they are solutions to the equation.

(36,1)

y = [tex]\frac{1}{4}[/tex]x - 8  Substitute in 36 for x and 1 for y

1 = [tex]\frac{1}{4}[/tex](36) - 8

1 = 9 - 8

1 = 1

You can do the same thing for all of the other points on the table.

Helping in the name of Jesus.

Answer:

x=28

Step-by-step explanation:

solve for x by simplifying both sides of the equation, then isolating the variable.

have a good day :)

Answer

x=3 1/9

Step-by-step explanation:

a. The perimeter of the paddock is 120 meters

b. The cost of the wire will be $600

a. To find the missing sides and calculate the perimeter of the paddock, let's analyze the given sketch.

The length of the missing side on the left can be determined by subtracting the known length of 10m from the total height of 20m. Therefore, the left side has a length:

20m - 10m = 10m.

The missing side on the right is equal in length to the known side of 40m.

Now we can calculate the perimeter by adding up all the side lengths:

Perimeter = 10m + 40m + 20m + 10m + 40m

= 120m.

b. To calculate the cost of the wire, we need to multiply the perimeter of the paddock by the cost per meter of fencing wire, which is $5/m.

Wire Cost = Perimeter × Cost per meter

= 120m × $5/m

= $600.

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Answer: Therefore, the first person gets $476,190.48 and the second person gets $323,809.52.

Step-by-step explanation: To divide $800,000 into a 25:17 ratio, we first need to add the ratio terms (25 + 17 = 42) to determine the total number of parts. Then, we divide the total amount by the total number of parts to determine the value of each part. Finally, we multiply the value of each part by the respective ratio term to determine the amount that each person gets.

The calculation steps are as follows:

Determine the total number of parts: 25 + 17 = 42

Determine the value of each part:

Value of each part = Total amount / Total number of parts

= $800,000 / 42

= $19,047.62 (rounded to two decimal places)

Determine the amount that each person gets by multiplying the value of each part by the respective ratio term:

First person gets: 25 parts * $19,047.62 per part = $476,190.48

Second person gets: 17 parts * $19,047.62 per part = $323,809.52

For the given right-angled triangle the measure of the angle is 11.30°.

Given adjacent base of the triangle =  10

given opposite side of the triangle = 2

To find out the angle we can use a little bit of trigonometry. By trigonometry, we know that tan ∠ = opposite side / adjacent side.

So, we can use the above tan ∠ relation to find the angle.

Tan ∠ = opposite side / adjacent side

Tan ∠ = 2/10

Tan ∠ = 1/5

∠ = Tan⁻¹(1/5)

∠ = 11.30°.

From the above explanation, we can conclude that the measure of the angle for the given right triangle is 11.30°.

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The missing probability is P(A or B) = 131/200

It should be noted that in order to find the probability of the union of two events A and B, i.e., P(A or B), we can use the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) = P(A) + P(B) - P(A and B) = (9/20) + (1/4) - (9/200)

Simplifying the expression, we get:

P(A or B) = 45/100 + 25/100 - 9/200 = (90+50-9)/200 = 131/200

The probability is 131 / 200.

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The number of games that the team did play in the last season is 23.

The frequency is the number of times an event happens. Therefore, frequency is the number of repetitions of a digit or an event.

The given frequency table represents the frequency of different runs scored by the team in the entire season.

The frequency is written in tally form, which means that each of the vertical lines represents the count of 1, while a group of fours lines such that the four lines are vertical while the fifth line cuts the other four represents a count of 5 as shown in the second column of the second row.

Therefore, the frequency table can be made as:

Number of runs     Frequency

            0                         8

            1                          5

            2                         3

            3                         6

            4                         1

Total:                              23

Hence, the team played 23 games.

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The linear function is the more appropriate function that has to be used here.

To determine which type of function better models the data, we can analyze the difference in the y values (Estimated Population) as x increases.

Differences between consecutive y values:

77,212 - 66,118 = 11,094

88,340 - 77,212 = 11,128

99,535 - 88,340 = 11,195

The differences between consecutive y values are relatively constant as x increases.

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You have to invest $15,506.08

PV = FV / (1 + r)^n

where:

PV = present value (amount of money you need to invest today)

FV = future value (amount of money you want to have in 5 years, which is $25,000)

r = interest rate (10% or 0.1)

n = number of periods (5 years)

Plugging in the values into the formula:

PV = $25,000 / (1 + 0.1)^5

PV = $25,000 / 1.61051

PV = $15,506.08

Therefore, you need to invest approximately $15,506.08 today at a 10% interest rate in order to have $25,000 in 5 years to purchase a car in cash.

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Answer:

the area of the shape is 48m

Step-by-step explanation:

10*4=40

2*4=8

40+8=48

brake it up into two triangles. one is a triangle that is 10m tall and 8m wide at the top. you can solve that by doing 4*10=40. the other triangle is 4m wide at the bottom and 2m tall. you can solve that by doing 2*4=8. and 40+8=48.

The constant of variation, k, is defined as the ratio between the two variables' values, multiplied by their respective powers:  [tex]k = y/x^a \times z/x^b,[/tex] where a and b are the powers of x in the expressions for y and z, respectively. when   [tex]x = -3[/tex]  and [tex]y = 4[/tex] , z is equal to -81/16.

In this case, we have x = 8, y = 54, and z = 144, so we need to determine the powers of x in the expressions for y and z:

[tex]y = 54 = (8^a) \times k = > a = 3[/tex]

[tex]z = 144 = (8^b) \times k = > b = 2[/tex]

Substituting these values into the formula for k, we get:

[tex]k = y/x^a \times z/x^b = 54/8^3 \times 144/8^2 = 27/64[/tex]

Therefore, the constant of variation is 27/64.

Using the same formula as above, we can solve for z when x = -3 and y = 4:

[tex]k = y/x^a \times z/x^b[/tex]

We need to determine the powers of x in the expressions for y and z:

[tex]y = 4 = (-3)^a * k = > a = -1[/tex]

[tex]z = ? = (-3)^b * k = >[/tex] we don't know b or z yet

Substituting these values into the formula for k, we get:

[tex]k = y/x^a * z/x^b = 4/(-3)^(-1) * z/(-3)^b = -12z/(-3)^2[/tex]

Simplifying this expression, we get:

[tex]k = -12z/9 = -4z/3[/tex]

Now we can solve for z:

[tex]-4z/3 = 27/64[/tex]

Multiplying both sides by 3/(-4), we get:

[tex]z = -81/16[/tex]

Therefore, when  [tex]x = -3[/tex] and   [tex]y = 4, z[/tex] is equal to   [tex]-81/16.[/tex]

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If a, b, c are in Harmonic Progression, b+ca+ca+b ac' ab bc are in Arithmetic Progression.

Given a, b, c are in H.P.:

1/a + 1/b = 2/c

Multiplying both sides by abc:

bc + ac = 2ab

Adding c to both sides:

bc + ac + c = 2ab + c

Rearranging the terms:

c + ab = ac + bc

Adding the terms b + ca and ca + b to both sides:

b + ca + ca + b + ac' + ab + bc = ac + bc + b + ca + ca + b

Simplifying:

b + ca + ca + b + ac' + ab + bc = 2ac + 2b

Dividing both sides by 2:

(b + ca + ca + b + ac' + ab + bc)/2 = ac + b

Hence, b+ca+ca+b ac' ab bc are in A.P.

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Answer:

Step-by-step explanation:

The standard form equation for an ellipse with center at the origin is:

x^2/a^2 + y^2/b^2 = 1

where 'a' is the distance from the center to the vertices along the x-axis and 'b' is the distance from the center to the vertices along the y-axis.

In this case, the center is at the origin, and a vertex is at (3, 0). So, we know that 'a' = 3.

We also know that the distance from the center to the focus is 'c', and that:

c^2 = a^2 - b^2

Since the center is at the origin, 'c' is the distance from the focus (1, 0) to the origin, which is 1. So, we can solve for 'b' as:

c^2 = a^2 - b^2

1^2 = 3^2 - b^2

b^2 = 3^2 - 1^2

b^2 = 8

b = sqrt(8) = 2sqrt(2)

Substituting these values into the standard form equation, we get:

x^2/3^2 + y^2/(2sqrt(2))^2 = 1

Simplifying:

x^2/9 + y^2/8 = 1

So, the equation in standard form for the given ellipse is:

9x^2 + 8y^2 = 72

The limit of the function will be:

lim f(x) = ∞ as x → ∞

lim f(x) = 0 as x → -∞

The limit of f(x) as x goes towards infinity is infinite while the limit of f(x) approaches 0 when x tends to a negative value. Mathematically speaking,

lim f(x) = ∞ as x → ∞

lim f(x) = 0 as x → -∞

We can comprehend such behavior because the function's base is greater than 1 which explains how its growth accelerates as x increases and diminishes quickly as x declines towards zero.

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Answer:

Step-by-step explanation:

The calculated value of the median of the A-list 7 members at the gym is 54

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

10, 64, 52, 46,54,67,54.

When the numbers are sorted in ascending order, we have

Sorted list = 10, 46, 52, 54, 54, 64, 67

The median is the middle number of the sorted list

So, we have

Middle number = 54

Thsis means that

Memdian = 54

Hence. the median of the A-list 7 members at the gym is 54

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