The length of a rectangle is 6 feet less than 4 times the width. If the perimeter is 68 feet, find the length and the width of the rectangle.

Answer:

  EF = 7

Step-by-step explanation:

You want to know the length of midsegment EF in the triangle with it labeled 43-4x and the parallel base segment labeled 32-2x.

The midsegment is half the length of the base segment, so we have ...

  (43 -4x) = 1/2(32 -2x)

  43 -4x = 16 -x . . . . . simplify

  27 = 3x . . . . . . . . add 4x-16

  9 = x . . . . . . . . divide by 3

  EF = 43 -4x = 43 -4(9) = 43 -36

  EF = 7

The derivative formula for the division of two functions is equal to d [θ(x)] / dx = [g(x) · f'(x) - f(x) · g'(x)] / [g(x)]², where θ(x) = f(x) / g(x).

In this problem we need to derive the formula for the derivative of the division of two functions, that is, the expression θ(x) = f(x) / g(x). First, we write the definition of the derivative for the given expression:

[tex]\frac{d}{dx} [\theta (x)] = \lim_{h \to 0} \frac{\theta(x + h) - \theta (x)}{h}[/tex]

Second, substitute and expand the expression by algebra properties:

[tex]\frac{d}{dx} [\theta (x)] = \lim_{h \to 0} \frac{\frac{f(x + h)}{g(x + h)} - \frac{f(x)}{g(x)} }{h}[/tex]

[tex]\frac{d}{dx}[\theta (x)] = \lim_{h \to 0} \frac{f(x + h) \cdot g(x) - f(x) \cdot g(x + h)}{h\cdot g(x + h)\cdot g(x)}[/tex]

Third, expand and simplify the expression one more time by algebra properties and limit properties:

[tex]\frac{d}{dx} [\theta (x)] = \lim_{h \to 0} \frac{f(x + h) \cdot g(x) - f(x) \cdot g(x) + g(x) \cdot f(x) -f(x) \cdot g(x + h)}{h \cdot g(x + h) \cdot g(x)}[/tex]

[tex]\frac{d}{dx}[\theta (x)] = \lim_{h \to 0} g(x) \cdot \frac{f(x + h) - f(x)}{h\cdot g(x+h)\cdot g(x)} - \lim_{h \to 0} f(x) \cdot \frac{g(x + h) - g(x)}{h\cdot g(x+h)\cdot g(x)}[/tex]

[tex]\frac{d}{dx}[\theta (x)] = \lim_{h \to 0} \frac{g(x)}{g(x + h) \cdot g(x)} \cdot \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} - \lim_{h \to 0} \frac{f(x)}{g(x+ h) \cdot g(x)} \cdot \lim_{h \to 0} \cdot \frac{g(x + h) - g(x)}{h}[/tex]

Fourth, simplify the resulting expression by evaluating the limits:

d [θ(x)] / dx = [g(x) / [g(x)]²] · f'(x) - [f(x) / [g(x)]²] · g'(x)

d [θ(x)] / dx = [g(x) · f'(x) - f(x) · g'(x)] / [g(x)]²

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Answer:13

Step-by-step explanation:

r = circumference/2pi

81.64/6.28 = 13

The radius of the garden is 13 units.

The solution to the complex number in rectangular form is; -2 + 2i

The complex number is given as;

2√2(cos 135° + i sin 135°)

The standard form is a + bi.

The value of cos 135° is; -(√2)/2

The value of sin 135° is; (√2)/2

Plugging these values into the given equation is;

2√2(-(√2)/2 + ((√2)/2)i)

Expanding the bracket gives;

-2 + 2i

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This student would receive 10 bag of popcorns.

Generally speaking, a unit price can be defined as the price that is being charged by a seller for the sale of a single unit of product.

In order to determine the quantity of popcorn, we would make use of the information provided about the unit price for one bag of popcorn by using the following mathematical expression;

Number of bag of popcorns = Cost of one bag of popcorn/Unit price

Substituting the given parameters into the mathematical expression above, we have the following;

Number of bag of popcorns = 5/0.50

Number of bag of popcorns = 10 bag of popcorns.

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The x- and y-components of the vector [tex]\mathbf{\hat a}}[/tex] are 8.9 m/s² and -6.4 m/s², respectively.

In mathematics, components are the parts or elements that make up a larger structure or object. In the context of vectors, components refer to the parts of a vector that describe its direction and magnitude in a particular coordinate system.

To find the x-components and y-components of the vector [tex]\mathbf{\hat a}}[/tex] = (11 m/s², 36 degrees left of -y-axis), we need to determine the direction of the vector. The notation "left of -y-axis" means that the vector makes an angle of 36 degrees with the negative y-axis, as measured counterclockwise from the negative y-axis.

The magnitude (or length) of the vector is given by the first component, which is 11 m/s².

The x-component of the vector can be found by multiplying the magnitude by the cosine of the angle:

x-component = magnitude x cos(angle) = 11 m/s² x cos(36°) = 8.9 m/s²

The y-component of the vector can be found by multiplying the magnitude by the sine of the angle:

y-component = magnitude x sin(angle) = 11 m/s² x sin(36°) = -6.4 m/s²

The negative sign indicates that the y-component is in the opposite direction of the negative y-axis.

Therefore, the x- and y-components of the vector [tex]\mathbf{\hat a}}[/tex] are 8.9 m/s² and -6.4 m/s², respectively.

Expressed as an ordered pair, the components are (8.9 m/s², -6.4 m/s²).

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a) A linear equation for the total remaining oil reserves, R, in terms of t, the number of years since the current year, is 2,090 - 18t.

b) Based on the above linear equation, 15 years from now, the total oil reserves will be 1,820 billion barrels.

c) Using division operation, the world's oil reserves will be completely depleted approximately 116 years from now.

A linear equation is a mathematical or algebraic equation written in the form of y = mx + b.

A linear equation consists of a constant and a first-order term, where m is the slope, and variable b, the y-intercept.

The current total world oil reserves, R = 2,090 billion barrels

Average decrease per year = 18 billion barrels

a) Linear equation for R is 2,090 - 18t

b) In 15 years, the total oil reserves = 2,090 - 18(15)

= 2,090 - 270

= 1,820

c) The constant rate of decrease = 18 billion barrels per year

The world's oil reserves will be completely depleted approximately 116 years from now (2,090/18).

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The number of pieces of candy that will be left on day 4 would be 2, 560 candies .

The number of pieces of candy left on day 8 would be 66 candies .

To find the number of candy left on the fourth day, given that Augustus only gives candies after the pile of 100, 000 candies is reached, the formula is:

= Number of starting candy x ( 1 - Percent given away ) ^ number of days

= 100, 000 x ( 1 - 60 % ) ⁴

= 100, 000 x 40 % ⁴

= 2, 560 candies

On Day 8, this would be:

= 100, 000 x ( 1 - 60 % ) ⁸

= 66 candies

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The question is:

How many pieces of candy will be left on day 4? On day 8?

Answer: The answer to your problem is choice b

Answer:

Step-by-step explanation:

Since Joe bought six bunches of seedless black grapes for $15, the cost per bunch of grapes is $15 / 6 = $2.50.

Since Anjali has $7.50, she can buy $7.50 / $2.50 = 3 bunches of seedless black grapes.

To find the weight of 2.2 cubic feet of water in pounds, we can use the following formula:

Weight in pounds = Volume in cubic feet x Density of water in pounds per cubic foot

The density of water is 62.4 pounds per cubic foot, so we can substitute the values into the formula:

Weight in pounds = 2.2 x 62.4 = 138.08

So, 2.2 cubic feet of water weighs 138.08 pounds.

To convert this weight to tons, we can divide by 2000, as there are 2000 pounds in a ton:

Weight in tons = 138.08 / 2000 = 0.06904 tons

So, 2.2 cubic feet of water weighs 0.06904 tons.

Answer:

Step-by-step explanation:

First, we need to find how many gallons of water 2.2 cubic feet hold:

2.2 cubic feet * 7.48 gallons per cubic foot = 16.456 gallons

Next, we'll find the weight of the water in pounds:

16.456 gallons * 8.33 pounds per gallon = 136.08 pounds

Finally, we'll convert the weight to tons:

136.08 pounds / 2000 pounds per ton = 0.0680 tons

So 2.2 cubic feet of water weigh 136.08 pounds and 0.0680 tons.

Answer:The range rule of thumb states that the standard deviation (s) is approximately equal to the range (R) divided by 4, where the range is the difference between the maximum and minimum values of a data set.

Using this formula, we can estimate the standard deviation of the heights as:

s = R / 4 = (192.5 - 152.0) / 4 = 20.625 / 4 = 5.156 cm

Comparing this estimate to the actual standard deviation of 12.22 cm, we see that the estimate is off by a factor of approximately 2.37.

Since the estimate is not within 1.7 cm of the actual standard deviation, it suggests that the range rule of thumb is not a very accurate way to estimate the standard deviation of this data set. The standard deviation calculated using the full set of data is much larger than the estimate, which means that the spread of the data is greater than what would be expected based on the range rule of thumb.

The linear functions for this problem are classified as follows:

a. Parallel.

b. Neither parallel nor perpendicular.

c. Perpendicular.

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

y = mx + b.

In which:

They are classified as parallel, perpendicular or neither according to their slopes as follows:

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Answer:

{40.7,49.3}

Step-by-step explanation:

[tex]\displaystyle CI=\bar{x}\pm z\frac{s}{\sqrt{n}}\\\\CI_{80\%}=45\pm1.282\biggr(\frac{19}{\sqrt{32}}\biggr)\\ \\CI_{80\%}\approx45\pm 4.3\\\\CI_{80\%}=\{40.7,49.3\}[/tex]

Thus, we are 80% confident that the true sample mean amount spent is between $40.70 and $49.30

a. The profit function is given by P(x) = R(x) - C(x). Substituting the expressions for R(x) and C(x), we have P(x) = 12x + 300√x - 2500.

b. The marginal profit function is the derivative of the profit function with respect to x. To find the marginal profit function, we will differentiate P(x) is dP/dx = 12 + 300/2√x.

c. The marginal average profit function is the derivative of the profit function divided by the number of units sold is dP/dx ÷ x = (12 + 300/2√x) ÷ x.

d. To estimate the profit from the sale of the 201st doll, we will evaluate the profit function at x = 201 is $16.62.

What do you mean by  profit function?

A profit function is a mathematical model that represents the profit made from selling a certain product or service as a function of the number of units sold. The profit function is calculated as the difference between the total revenue and the total cost of producing and selling the product.

Mathematically, if R(x) represents the total revenue from selling x units of a product and C(x) represents the total cost of producing and selling x units of the product, then the profit function can be represented as P(x) = R(x) - C(x).

a. The profit function is given by P(x) = R(x) - C(x). Substituting the expressions for R(x) and C(x), we have:

P(x) = 14x - (2x - 300√x + 2500)

P(x) = 12x + 300√x - 2500

b. The marginal profit function is the derivative of the profit function with respect to x. To find the marginal profit function, we will differentiate P(x):

dP/dx = 12 + 300/2√x

c. The marginal average profit function is the derivative of the profit function divided by the number of units sold.

dP/dx ÷ x = (12 + 300/2√x) ÷ x

d. To estimate the profit from the sale of the 201st doll, we will evaluate the profit function at x = 201:

P(201) = 12 * 201 + 300 * √201 - 2500 = $3347.09

Since we want the profit from the sale of one unit, we divide the profit by 201:

P(201) ÷ 201 = $16.62

So, the estimated profit from the sale of the 201st doll is $16.62.

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Answer:

[tex]x > 7[/tex]

Step-by-step explanation:

Given the inequality

[tex]x+4 > 11[/tex]

Lets solve the inequality for [tex]x[/tex].

Subtract [tex]4[/tex] from both sides of the inequality.

[tex]x > 11-4[/tex]

Evaluate [tex]11-4[/tex].

[tex]x > 7[/tex]

[tex]x[/tex] is greater than 7

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Answer:

A.

Step-by-step explanation:

Since [tex]5[/tex] can rewritten as [tex]\sqrt{25}[/tex] and [tex]6[/tex] rewritten as [tex]\sqrt{36},[/tex] we can see which options go between [tex]\sqrt{25}[/tex] and [tex]\sqrt{36}.[/tex] The only square root that goes between is [tex]\sqrt{29}.[/tex] Therefore the answer is [tex]\boxed{A.}[/tex] (or [tex]\sqrt{29}.[/tex])

Based on the proceeds and effective rate, Wilson should take out the simple discount note, which has a higher proceeds of $5,005 and a higher effective rate of 9.9% compared to the simple interest note's proceeds of $5,995 and effective rate of 9.0%.

To compare the simple discount note and the simple interest note, we can calculate the proceeds and effective rates for each.

For the simple discount note, the interest is deducted upfront and the borrower receives the difference, or the proceeds. The interest on a $5,500 note at 6% for 18 months can be calculated as:

Interest = Principal × Rate × Time

Interest = $5,500 × 6% × 1.5 years

Interest = $495

The proceeds for the simple discount note can be calculated as:

Proceeds = Principal - Interest

Proceeds = $5,500 - $495

Proceeds = $5,005

The effective rate for the simple discount note can be calculated using the formula:

Effective Rate = Interest / Proceeds × 100%

Effective Rate = $495 / $5,005 × 100%

Effective Rate = 9.9%

For the simple interest note, the interest is calculated on the principal amount and added to the principal at the end of the term. The interest on a $5,500 note at 6% for 18 months can be calculated as:

Interest = Principal × Rate × Time

Interest = $5,500 × 6% × 1.5 years

Interest = $495

The total amount due at the end of the term for the simple interest note can be calculated as:

Total Due = Principal + Interest

Total Due = $5,500 + $495

Total Due = $5,995

The effective rate for the simple interest note can be calculated using the formula:

Effective Rate = Interest / Principal × 100%

Effective Rate = $495 / $5,500 × 100%

Effective Rate = 9.0%

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Answer:

27,64,216,64,243

Step-by-step explanation:

Any number raise to power 3

The probability that more than 2 students will attend is 0.9179.

Part (a) List the values that X may take on.

X may take on the values 0, 1, 2, 3, ..., 13.

Part (b) Give the distribution of X.

The distribution of X can be represented as follows: X-0:0, 1:0.23, 2:0.23, 3:0.23, 4:0.23, 5:0.23, 6:0.23, 7:0.23, 8:0.23, 9:0.23, 10:0.23, 11:0.23, 12:0.23, 13:0.23

Part (c) How many of the 13 students do we expect to attend the festivities? (Round your answer to the nearest whole number.)

We can expect 3 students to attend the festivities.

Part (d) Find the probability that at most 3 students will attend. (Round your answer to four decimal places.)

The probability that at most 3 students will attend is 0.6923.

Part (e) Find the probability that more than 2 students will attend. (Round your answer to four decimal places.)

The probability that more than 2 students will attend is 0.9179.

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The volume of the parallelopiped is 4

We have to solve this with the use of matrix system in math

PQ, PR, PS where

P(-3, 3, 2), Q(-1, 6, 5), R(-4, 2, 1), S(3, 1, 4)

we have to find

Q - P

(-1, 6, 5) - (-3, 3, 2)

= (2, 3, 3)

R - P

R(-4, 2, 1) - (-3, 3, 2)

= ( -1 , -1, -1)

S - P

S(3, 1, 4) - (-3, 3, 2)

= (6 , -2, 2)

we would have [tex]\left[\begin{array}{ccc}2&3&3\\-1&-1&-1\\6&-2&2\end{array}\right][/tex]

the determinant of the matrix is the volume

2[(-1 * 2) - 3[(-1x2)-(6x-1)] + 3[(-1x-2) -(6 x -1)]

= -8 - 12 + 24

= 4

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Answer:

Step-by-step explanation:

C. There is a moderate correlation between the scores and grades.

Based on the scatter plot and the calculated correlation coefficient, we can conclude that there is a moderate positive correlation between the pretest scores and the course grades. This means that as the pretest scores increase, the course grades tend to increase as well. However, the correlation is not extremely strong, so other factors may be affecting the course grades besides the pretest scores.

The correlation between two variables describes the relationship between them. The strength of the correlation can be classified into four categories: no correlation, weak correlation, moderate correlation, and strong correlation.

In this case, since the values of X and Y appear to have a moderate association, it might be concluded that there is a moderate correlation between the scores on the pretest and the course grade.

However, correlation does not imply causality, meaning that even if there is a correlation between two variables, it does not mean that one variable causes the other. Further analysis, such as regression analysis, is needed to determine the strength and direction of the relationship and to determine if there is a causal relationship.

The equation y² - y²x = 6  can be describe as a function of x as:

y = √(6/(x - x)).

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

y² - y²x = 6

Taking y² as common.

y²(1 - x) = 6

Divide both sides with (1 - x).

y² = 6/(1 - x)

Square root on both sides.

y = √(6/(x - x))

This is a function of x.

Thus,

y = √(6/(x - x)) is a function of x.

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Answer:

Step-by-step explanation:

m=3-1/2-7= -2/5

c⇒ -2

y=mx+c

y=-2/5x-2

Answer:

To find the probability of Miko taking a red marble first, then a white marble, we can use the formula for joint probability:

P(red, then white) = P(red) * P(white | red)

Where P(red) is the probability of Miko taking a red marble first and P(white | red) is the probability of Miko taking a white marble second, given that he took a red marble first.

P(red) = 3/12 = 1/4

P(white | red) = 4/12 = 1/3

So,

P(red, then white) = 1/4 * 1/3 = 1/12

Therefore, the probability of Miko taking a red marble first, then a white marble is 1/12.

Step-by-step explanation:

(a) To convert feet to fathoms, we can use the conversion factor 1 fathom = 6 feet. Hence, we can divide the depth in feet by 6 to find the depth in fathoms:

36,078 feet / 6 feet/fathom = 6,013 fathoms

So the depth of the seabed is 6,013 fathoms.

(b) To convert feet to leagues (marine), we can use the conversion factor 1 league (marine) = 5,556 feet. Hence, we can divide the depth in feet by 5,556 to find the depth in leagues (marine):

36,078 feet / 5,556 feet/league (marine) = 6.52 leagues (marine)

So the depth of the seabed is approximately 6.52 leagues (marine).

The scaled dimensions of a kitchen with dimensions 6.3m by 4.8m, if the kitchen plan is drawn to a scale of 1:500, are given as follows:

1.26 cm and 0.96 cm.

The scaled dimensions are obtained applying the proportions in the context of the problem.

The scale is of 1:500, meaning that each dimension is divided by 500, as follows:

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1. The amount of pizzas which the pizzeria must sell for employees to receive the bonus is 78.

2. The new number of pizzas sold is considered as percent increase of 20%.

The pizzeria sold 65 pizzas yesterday and the owner offers bonus if the number sold increases by 20% today. The number that must be sold, to receive a bonus is computed as follows:

= 65 pizza * (65 pizza * 20%)

= 65 pizza * 13 pizza

= 78 pizza.

It is a percent increase which is computed as follows. The formula for percentage increase is: ((Final value - Initial value)/Initial value * 100).

The Percent increase:

= (78 pizza - 65 pizza) / 65 pizza.

= 13 pizza / 65 pizza

= 0.2

= 20%

Full question "A local pizzeria sold 65 pizzas yesterday. (step 1) The owner offers the employees a bonus if the number of pizzas sold increases by 20% today. How many pizzas must the pizzeria sell for employees to receive the bonus? Explain your reasoning. (Part 2) Suppose the actual number of pizzas sold today was 60. Describe the number of pizzas sold as a percent increase or decrease, rounded to the nearest hundredth. Show your work. Answer:"

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