Express log 16+ log 4 in terms of log 2

Answer:

cost of wristband = $7.00

cost of each ride = $2.50

Step-by-step explanation:

w = wristband fee

r = cost of one ride

(1)  w + 3r = 14.50

(2) w + 6r = 22.00

Solve equation (1) for w, then substitute into equation (2):

w = 14.50 - 3r

(14.50 - 3r) + 6r = 22.00   now solve for r

3r = 22 - 14.5 = 7.5

r = 7.5/3 = 2.5  now solve for w

w = 14.5 - 3(2.5) = 14.5 - 7.5 = 7

False.

If the linear transformation T(x) = Ax is onto (also known as surjective), it means that for every vector y in the output space of T, there exists at least one vector x in the input space of T such that T(x) = y.

If the linear transformation T(x) = Ax is onto (also known as surjective), it means that for every vector y in the output space of T, there exists at least one vector x in the input space of T such that T(x) = y. However, this does not necessarily imply that T(x) is one-to-one (also known as injective).

To see why, consider a simple example where A is a 2x2 matrix such that A = [1 0; 0 0]. The linear transformation T(x) = Ax maps a 2-dimensional vector x to a 2-dimensional vector Ax, and its matrix representation is given by:

| 1 0 |
| 0 0 |

This transformation is onto, since every vector y in the output space of T can be written as y = Ax for some vector x in the input space of T. However, T(x) is not one-to-one, since T([1 1]) = T([2 0]) = [1 0], which means that two different input vectors map to the same output vector.

Therefore, the statement "if the linear transformation T(x) = Ax is onto, then it is also one-to-one" is false. In fact, a linear transformation is one-to-one if and only if its null space (also known as kernel) is trivial, which means that the only vector that maps to the zero vector is the zero vector itself.

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The equation for the circle with a center at (-2,5) and a radius of 3 is (x + 2)² + (y - 5)² = 9.

Given that:

Center, (h, k) = (-2, 5)

Radius, r = 3

Let r be the radius of the circle and the location of the center of the circle be (h, k). Then the equation of the circle is given as,

(x - h)² + (y - k)² = r²

The equation for the circle with a center at (-2,5) and a radius of 3 is given as,

(x + 2)² + (y - 5)² = 3²

(x + 2)² + (y - 5)² = 9

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The p-value of 0.6901, we did not find enough evidence to say a significant difference exists between the true average fuel economy today and 22.74 MPG (option 5).

Based on the given information, the null hypothesis is that the true average fuel economy today is equal to 22.74 MPG, while the alternative hypothesis is that it is not equal to 22.74 MPG. The p-value of 0.6901 indicates that there is a 69.01% chance of obtaining the observed sample mean or one more extreme, assuming the null hypothesis is true.

Since the p-value is greater than the significance level of 5%, we fail to reject the null hypothesis. This means that we do not have enough evidence to say that the true average fuel economy today is significantly different from 22.74 MPG.

Therefore, the appropriate conclusion would be option 5: We did not find enough evidence to say a significant difference exists between the true average fuel economy today and 22.74 MPG.

In this scenario, we are conducting a one-sample mean hypothesis test to determine whether the average fuel economy today is different from 22.74 MPG. The null hypothesis (u = 22.74) states that there is no significant difference, while the alternative hypothesis (u ≠ 22.74) states that there is a significant difference.

We are using a 5% level of significance to make our decision. The p-value observed in this test is 0.6901, which is greater than the significance level of 0.05. Therefore, we do not have enough evidence to reject the null hypothesis.

In conclusion, based on the given information and the p-value of 0.6901, we did not find enough evidence to say a significant difference exists between the true average fuel economy today and 22.74 MPG (option 5).

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Answer:

Perimeter = 48 inches

Area = 144 inches²

Step-by-step explanation:

Perimeter = 4 * side

Perimeter = 4*12 in

Perimeter = 48 inches

Area = side²

Area = (12in)²

Area = 144 in²

The rectangle that has side lengths of 5 units and 4 units is C) A(3,3), B(3,7), C(7,7), D(7,3).

Point C at (7,7), means the width is 4 units (7-3) and the height is 5 units (7-2), so this is the correct rectangle.

A rectangle is a shape with four right angles (that is four angles of 90 degrees) and the opposite sides are parallel and congruent.

The two sides of a rectangle are parallel and they meet at the four corners or vertices.

For a rectangle, the opposite sides are of  the same length and are parallel to each other.

Therefore, C. A(3,3), B(3,7), C(7,7), D(7,3) is the rectangle that has side lengths of 5 units and 4 units.

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The graphical solution of the quadratic equation, x² = 6·x - 9 is; x = 3

A quadratic equation is an equation that can be expressed in the form; f(x) = a·x² + b·x + c, where a ≠ 0, and a, b, c are numbers.

The graphical solution of the equation x² = 6·x - 9, which is a quadratic equation can be found by graphing the expressions, x² and 6·x - 9 on the same coordinate plane.

The coordinates of the points on the expression; x² are as follows;

(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), and (5, 25), (6, 36), (7, 49), (8, 64), (9, 81), (10, 100),

The coordinates of the points on the expression; 6·x - 9 are as follows;

(0, -9), (1, -3), (2, 3), (3, 9), (4, 15), and (5, 21)

The above points indicates that the solution of the equation, x² = 6·x - 9, obtained by comparing the points to be graphed is the point (3, 9)

Please find attached the graph of the specified quadratic equation, showing the solution point, created with MS Excel

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The calculated volume of the sphere is 50939.2 cubic foot

Given that the radius of the sphere is represented as

Radius, r = 23 ft

The formula of the volume of the sphere is represented as

V = 4/3πr³

Where

By substitution, we have

V = 4/3 * 3.14 * 23^3

Evaluate the products in the above equation

V = 50939.2 cubic foot

Hence, the calculated volume of the sphere is 50939.2 cubic foot

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Let's try to understand how we can solve this equation

Given equation,4f+2=6f-12

Now we will take the terms on one side and constants on the other.

=> 2+12=6f-4f

=> 14=2f

=>7=f

So, the value of f would be 7. Remember that while bringing value to another side, the sign of that value changes. If it is a positive sign then it is going to be transformed into a negative one and vice versa.

The probability of the spinner first not landing on green, spun again, and then landing on green is P ( A ) = 0.2475

Given data ,

Let the probability of the spinner first not landing on green, spun again, and then landing on green is P ( A )

Now , Since the likelihood of the spinner landing on green on each spin is independent and constant, the chance that it will do so on the second spin is 0.45.

We add the probabilities together to determine the likelihood that the spinner won't land on green at first, then will spin again and land on green.

On the first spin, there is a 0.55 percent chance of not landing on green.

The second spin's probability of landing on green is 0.45.

Probability of landing on green after not landing on green is equal to 0.55 times 0.45.

P ( A ) = 0.2475

Hence , the probability that the spinner will first miss landing on green, spin again, and finally land on green is 0.2475, or 24.75%

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The formula you provided seems to have a formatting issue, so I'll assume you're asking about the sample size formula for a two-tailed hypothesis test. In that case, the formula used is: n = (Z_α/2 * σ / E)^2

The formula you have provided is used to calculate the required sample size (n) for a two-tailed hypothesis test when the level of significance (α) and the margin of error (E) are known. The formula is not a derivative of any other formula but is derived using the standard normal distribution and the formula for the margin of error.

The formula is: n = (Za/2)^2 * p(1-p) / E^2

where Za/2 is the critical value of the standard normal distribution for a two-tailed test at level of significance α/2, p is the estimated proportion of the population with the characteristic of interest, and E is the margin of error.

In your specific case, the formula is:

n = 12 * (0.2/2 + Z0.05)^2 / (0.04)^2

where α = 0.05 (two-tailed test at 95% confidence level), p = 0.2 (estimated proportion of the population with the characteristic of interest), and E = 0.04 (margin of error).

However, the value of Za/2 is not given in the formula you provided, so it is not possible to calculate the required sample size without additional information.

The formula you provided seems to have a formatting issue, so I'll assume you're asking about the sample size formula for a two-tailed hypothesis test. In that case, the formula used is:

n = (Z_α/2 * σ / E)^2

Here, n represents the required sample size, Z_α/2 is the Z-score for the desired level of confidence (α/2), σ is the population standard deviation, and E is the margin of error.

We use this formula when we want to determine the sample size required to estimate a population parameter (like the mean) within a specific margin of error while considering a certain level of confidence. It is derived from the general formula for calculating the margin of error in a hypothesis test.

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The solid obtained by rotating the region bounded by the curves y = x(x – 1)² and y = 0 about the y-axis, the volume V of solid S is: V = (π/15) cubic units

To find the volume of S using cylindrical shells, we can integrate the area of a cylindrical shell over the interval [0,1], where the radius of the shell is x and the height is given by the difference between the y-values of the two curves.

The height of the shell is y = x(x-1)², and the radius is x. The circumference of the shell is 2πx. Therefore, the volume of the shell is:

dV = 2πxy dx
dV = 2πx(x-1)² dx

To find the total volume, we integrate over the interval [0,1]:

V = ∫₀¹ 2πx(x-1)² dx

Using integration by substitution, we can simplify this integral as follows:

Let u = x-1, then du = dx

V = ∫₋₁⁰ 2π(u+1)u² du
V = ∫₋₁⁰ (2πu³ + 2πu²) du
V = [πu⁴ + 2/3πu³] from u = -1 to u = 0
V = π(0⁴ + 2/3(0³) - (-1)⁴ - 2/3(-1)³)
V = π(1 + 2/3)
V = 5π/3

Therefore, the volume of the solid S is 5π/3 cubic units.

To find the volume V of solid S obtained by rotating the region bounded by the curves y = x(x-1)² and y = 0 about the y-axis, we'll use the cylindrical shells method.

The formula for the volume using cylindrical shells is:
V = 2π * ∫[a, b] (radius * height) dx

In this case, the radius is x and the height is x(x-1)². The limits of integration can be found by setting y = 0:
0 = x(x-1)²
This yields x = 0 and x = 1.

So, the volume V can be found using:
V = 2π * ∫[0, 1] (x * x(x-1)²) dx

Now, we evaluate the integral:
V = 2π * ∫[0, 1] (x²(x-1)²) dx

By solving this integral, we get:
V = 2π * (1/30)

So, the volume V of solid S is:
V = (π/15) cubic units

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a) The function that has a greater b value is given as follows: Function B.

b) Both functions have an horizontal asymptote at y = 0.

An exponential function has the definition presented as follows:

y = ab^x.

In which the parameters are given as follows:

For Function B, we have that when x increases by one, y is multiplied by a value greater than 3, as:

Hence function B has a greater b-value.

Both functions have an horizontal asymptote at y = 0, as we can see from the graph of function B, as well as from the fact that there is no adding/subtracting term in function A.

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The equation of the line passing through points (3, 17) and (6, 32) is y = 5x + 2.

An equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of all the points on a straight line.

It takes the form y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line intersects the y-axis). This equation allows us to easily calculate the y-coordinate of any point on the line, given its x-coordinate.

The equation of the line is,

y =mx+c

The slope of the line is,

m = (y₂-y₁) / ( x₂ - x₁)

m = ( 32-17) / (6-3)

m = 15/3

m = 5

The y-intercept of the line is,

y = mx + c

17 = 5 x 3 + c

c = 2

The equation of the line is,

y = mx + c

y = 5x + 2

The graph of the line is attached with the answer below.

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The value of the diameter of the cone part of the funnel to the nearest tenth is,

⇒ d = 7.2 cm

We have to given that;

A funnel can hold 159π cm³ of fluid.

And, Its height (without the stem) is 12 cm.

Now, We know that;

Volume of cone = πr²h/3

Where, r is radius of cone.

Hence we get;

⇒ 159 = 3.14 × r² × 12 / 3

⇒ 159 = 12.56 r²

⇒ r² = 12.65

⇒ r = √12.65

⇒ r = 3.556

⇒ r = 3.6 cm

Thus, The value of the diameter of the cone part of the funnel to the nearest tenth is,

⇒ d = 2r

⇒ d = 2 x 3.6

⇒ d = 7.2 cm

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Answer:

missing angle measure = 100°

Step-by-step explanation:

the sum of the interior angles of a polygon is

sum = 180° (n - 2 ) ← n is the number of sides

a pentagon has n = 5 , then

sum = 180° × (5 - 2) = 180° × 3 = 540°

given sum of 4 angles = 440° , then

missing angle = 540° - 440° = 100°

The average price of milk in 2018 was 6.45 dollars per gallon.

The average price of milk in 2021 was 189.95 dollars per gallon.

The given function is:

Price = 3.55 + 2.90(1 + x)³

where x is the number of years after 2018.

In order to find the average price of milk in 2018, we need to set x = 0:

Price in 2018 = 3.55 + 2.90(1 + 0)³ = 3.55 + 2.90(1) = 6.45 dollars per gallon

Price in 2021 = 3.55 + 2.90(1 + 3)³ = 3.55 + 2.90(64) = 189.95 dollars per gallon

So, the average price of milk in 2021 was 189.95 dollars per gallon.

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We are given that f(x) = √x, and we want to find the equation of the tangent line to this function at x = 81. We can use the formula for the equation of the tangent line:

y - f(a) = f'(a)(x - a),

where f'(a) is the derivative of f(x) evaluated at x = a.

First, we calculate the derivative of f(x) as:

f'(x) = 1/(2√x)

Evaluated at x = 81, we get:

f'(81) = 1/(2√81) = 1/18

So the equation of the tangent line to f(x) at x = 81 is:

y - √81 = (1/18)(x - 81)

Simplifying:

y - 9 = (1/18)x - (1/2)

y = (1/18)x + 8.5

Now we can use this equation to approximate √81.1:

√81.1 ≈ (1/18)(81.1) + 8.5

≈ 9.0139

Therefore, using linear approximation with the tangent line, we can approximate √81.1 as 9.0139.

Marko has made profit of 17 dollars.

Given that, Marko earns $15 for each lawn he mows, $10 for each yard he weeds, and $5 for each yard he rakes.

Total amount spent = $(3+36+14)

= $53

Total money earned = $(15×3+5×5)

= $(45+25)

= $70

Profit = Total money earned-Total amount spent

= 70-53

= $17

Therefore, Marko has made profit of 17 dollars.

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The missing numbers in the equations when completed are  (-7) x 2 = -14 and 5 x 3 = -15

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

(-7) x ? = -14 ? x 3 = -15

Solving the equations, we have

(-7) x ? = -14

Divide both sides by -7

? = 2

Next, we have

? x 3 = -15

Divide both sides by 3

? = -5

Hence, the solutions are 2 and -5

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Therefore, the probability that exactly 5 of the students in a study group of 10 have studied in the last week is approximately 0.2461.

To calculate the probability that exactly 5 of the students in a study group of 10 have studied in the last week, we need to use the binomial distribution formula:

[tex]P(X=k) = C(n,k) * p^k * (1-p)^{(n-k)}[/tex]

where:

P(X=k) is the probability that k students have studied in the last week

n is the total number of students in the group (n = 10)

k is the number of students who have studied in the last week (k = 5)

p is the probability that a student has studied in the last week

Since we don't have information about p, let's assume that it's 0.5, which means that each student has an equal chance of having studied in the last week or not.

Using this assumption and plugging in the values, we get:

[tex]P(X=5) = C(10,5) * 0.5^5 * 0.5^{(10-5)[/tex]

= 252 * 0.03125 * 0.03125

= 0.2461

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The steady-state pmf of the Markov chain: π₀ ≈ 0.48, π₁ ≈ 0.32, π₂ ≈ 0.20

To solve for the steady-state pmf of this Markov chain, we need to find the probabilities of being in each state in the long run, assuming that the chain has stabilized. We can do this by solving the system of equations:

π = πP

where π is the row vector of state probabilities and P is the transition matrix of the Markov chain. In this case, the transition matrix is:

P =
0.6 0.7 0.4
0.1 0.1 0.8
0   0.2 0.3

and the initial state probabilities are:

π = (1 0 0)

Substituting these values into the equation, we get:

π = πP
(1 0 0) = (1 0 0)P
1 = 0.6π1 + 0.1π2
0 = 0.7π1 + 0.1π2 + 0.2π3
0 = 0.4π1 + 0.8π2 + 0.3π3

Simplifying these equations, we get:

π1 = 0.4π2
π3 = 2π2

Substituting these values back into the second equation, we get:

0 = 0.7π1 + 0.1π2 + 0.4π2
0 = 0.7π1 + 0.5π2
π2 = 1.4π1

Substituting these values into the first equation, we get:

1 = 0.6π1 + 0.1(1.4π1)
1 = 0.76π1
π1 = 1/0.76 ≈ 1.3158

Substituting this value back into the other equations, we get:

π2 ≈ 1.7368
π3 ≈ 3.4737

Therefore, the steady-state pmf of this Markov chain is:

π ≈ (0.4103 0.5789 0.0108)

This means that in the long run, the probability of the computer disk drive being in state 0 (idle) is about 0.41, the probability of being in state 1 (read) is about 0.58, and the probability of being in state 2 (write) is very low at about 0.01.

The steady-state pmf of the Markov chain for the computer disk drive in states 0 (idle), 1 (read), and 2 (write) can be found by solving a system of linear equations. Given the Markov chain transition probabilities:

0.6 0.7 0.4
0.1 0.1 0.6
0.3 0.2 0.0

Let π = [π₀, π₁, π₂] be the steady-state probabilities.

We have the following system of linear equations:

π₀ = 0.6π₀ + 0.1π₁ + 0.3π₂
π₁ = 0.7π₀ + 0.1π₁ + 0.2π₂
π₂ = 0.4π₀ + 0.6π₁ + 0.0π₂
π₀ + π₁ + π₂ = 1

Solving the system, we find the steady-state pmf of the Markov chain:

π₀ ≈ 0.48
π₁ ≈ 0.32
π₂ ≈ 0.20

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Answer:

2.9

Step-by-step explanation:

cos 70° = 0.342020

1/cos 70° = 1/0.342020 = 2.9238

Answer: 2.9

Answer:

3.16 x 10⁵ or 316,000

Step-by-step explanation:

We can start by dividing the mass of the sun by the combined mass of the moon, Earth, and Pluto:

1.898 x 10³⁰ / 5.986 x 10²⁴ ≈ 3.16 x 10⁵

This means that approximately 3.16 x 10⁵ (or 316,000) combined masses of the moon, Earth, and Pluto are needed to equal the mass of the sun.

Answer:

[tex]p(t)= 125.9( {.66}^{t} )[/tex]

t = number of years since 2014

p(t) = butterfly population (in millions)

[tex]p(5) = 125.9( {.66}^{5} ) = 15.8[/tex]

So the butterfly population in 2019 was about 15.8 million.

As per the concept of exponential equation, the population of the butterfly variety at the end of 2019 is approximately 33.02 million.

To calculate the population at the end of 2019, we need to apply the exponential decay formula. Exponential decay occurs when a quantity decreases over time at a constant percentage rate.

The formula for exponential decay is:

Population(t) = Population₀ * (1 - r)ᵃ

Where:

Population(t) is the population at time x.

Population₀ is the initial population (at t=0 or the given starting point).

r is the rate of decay (expressed as a decimal).

x is the time elapsed (in this case, the number of years from the starting point).

We are given that the rate of decrease is 34% per year, which can be written as 0.34 in decimal form. The population at the end of 2014 (t=0) is 125.9 million.

Using the formula, let's calculate the population at the end of 2019 (t=5 years from 2014):

Population(2019) = 125.9 million * (1 - 0.34)⁵

Population(2019) = 125.9 million * (0.66)⁵

Population(2019) = 125.9 million * 0.262144

Population(2019) ≈ 33.02 million

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Your answer: A  −14 × {23 + (−47)} = [−14 × 23] + [−14 × (−47)]  This example demonstrates the distributive property of multiplication over addition for rational numbers, which states that for any rational numbers a, b, and c: a × (b + c) = (a × b) + (a × c).

The correct answer is A: −14 × {23 + (−47)} = [−14 × 23] + [−14 × (−47)]. This is an example of the distributive property of multiplication over addition for rational numbers because we are multiplying −14 by the sum of 23 and −47, and we can distribute the multiplication to each term inside the parentheses by multiplying −14 by 23 and −14 by −47 separately, and then add the two results together. This is the basic definition of the distributive property of multiplication over addition. Option B shows the distributive property of multiplication over subtraction, option C shows the product of multiplication and addition, and option D is not a valid equation.

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Answer:

2

Step-by-step explanation:

75/8=5

(75/8)/5=5/5

3 3/4=2

Step-by-step explanation:

The cube has SIX sides ....each has area   x * x

  total area =   6    *  x*x   =   216     cm^2

                         6x^2 = 216

                              x^2 = 36

                               x = 6 cm

Volume =   x * x * x = 216 cm^3

To answer your question, we need to know some additional information such as the initial velocity of the ball and the equation of the parabolic path it follows. Without this information, we cannot determine the exact distance from the tree house or the height of the ball. However, we can make some assumptions based on typical scenarios.

Assuming that Emily threw the ball with a moderate force, we can estimate the distance from the tree house to be around 10-20 feet. The height of the ball would depend on how high the tree house is, but we can assume it is around 10-15 feet.

To find the distance from the tree house where the ball strikes the ground, we need to know the equation of the parabolic path. Without this information, we cannot determine the exact distance. However, we can estimate the range of the ball based on typical scenarios. Assuming that Emily threw the ball at an angle of around 45 degrees and with a moderate force, the ball would travel around 50-100 feet before hitting the ground.

In summary, without more information about the initial conditions and the equation of the parabolic path, we can only make estimates of the distance from the tree house and the height of the ball, and an educated guess about the distance the ball traveled before hitting the ground.

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The simple interest when the principal is $1,500, the interest rate is 6.0%, and the time is 5 years is $450.

To find the simple interest, you can use the formula:

Simple Interest (SI) = Principal (P) × Interest Rate (R) × Time (T)

Here, the principal (P) is $1,500, the interest rate (R) is 6.0%, and the time (T) is 5 years.

First, convert the interest rate from percentage to decimal by dividing by 100:

R = 6.0 / 100 = 0.06
 
Now, plug in the values into the formula:

SI = P × R × T
SI = $1,500 × 0.06 × 5

Calculate the result:

SI = $1,500 × 0.06 × 5 = $450

So, the simple interest for this case is $450.

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