Leaves and acorns are falling from the trees, but to work out the exact number of each you need to solve the following equation.

If l number of leaves = 46, what does a (acorns) equal?

l + 3a = 145

The equation most closely represents the line depicted in the graph is

f(x) = 60x − 160

An equation between two variables that gives a straight line when plotted on a graph.

Given a point (4,80) and a slope, m = 60, these can be used to formulate the equation of the graph. Using the point-slope formula:

y - y1 = m(x - x1)

y - 80 = 60(x - 4)

y - 80 = 60x - 240

y = 60x -160

Taking the x and y-intercepts:

At y = 0

0 = 60x - 160

x = 2.7

At x = 0

y = 60(0) - 160

y = -160

Hence, you could state that there is a graph of a line that has a y intercept that is approximately negative $160 million and an x intercept that is approximately 2.7 years.

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Answer:

Kendra will donate $9.86 to her Best Friend's Animal Society.

Step-by-step explanation:

In order to find this answer, we need to multiply $11.60 by the 8.5 hours:

11.60 x 8.5 = $98.60

Now, if she donates 1/10 every week, we can turn 1/10 into 0.10. And on a calculator set up this equation:

0.10 x 98.60 = $9.86

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The smallest number from the sum of three consecutive even numbers is 14.

Given, The sum of three consecutive even numbers is 48.

we are asked to determine the smallest number of the numbers.

Let:

x= the 1st con.even number

x+2=the 2nd con.even number

x+4=the 3rd con.even number

Add the terms and equate it with the total, 48

x+(x+2)+(x+4) = 48

simplify

x+x+2+x+4=48

combine the like terms.

3x+6=48

3x = 48-6

3x=42

x=42/3

x=14

the three numbers are:

x = 14

x+2 = 14+2 = 16

x+4 = 14+4 = 18

smallest number of the three numbers = 14

Hence we get the smallest number as 14.

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The length of the sides of the squares is 3 cm.

A rectangle is a quadrilateral with all four interior angles 90°.

Given that, the dimension of the rectangular tin piece is 40 cm by 50 cm.

Let the length of the sides of the square is x.

Then, the dimensions of the resulting rectangular base are (40 - 2x) by

(50 -2x).

Given that, the area of the resulting base is 1496 square cm.

Therefore, it follows:

(40 - 2x)(50 -2x) = 1496

2000 - 80x -100x + 4x² = 1496

4x² - 180x + 2000 -1496 = 0

4x² - 180x + 504 = 0

x² - 45x + 126 = 0

x² - 42x - 3x + 126 = 0

x(x- 42) - 3(x - 42) = 0

(x - 3)(x - 42) = 0

x-3 =0 or x - 42 = 0

x = 3 or x = 42

Note that the dimension of the rectangular tin piece is 40 cm by 50 cm, therefore, x = 42 cm cannot be the length of the square.

Hence, the length of the sides of the squares is 3 cm.

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a. Please find attached the sample space

b. P(sum of 8) = 5/36

c. P(odd sum or sum greater than 9) = 2/3

d. P(odd sum and sum greater than 9) =  1/12

e. P(A) = 1/3

f. P(B) = 1/6

g. P(A and B) = 1/18

h. A and B are not mutually exclusive because both events can occur at the same time.

i. A and B are not independent because the occurrence of A and lead to the occurrence of B.

P(sum of 8) = number of sample spaces that sum up to 8 / total number of sample spaces

P(sum of 8) = 5/36

P(odd sum or sum greater than 9) = (sample spaces that have an odd sum / total number of sample spaces) + (sample spaces that have a sum greater than 9 / total number of sample spaces)

P(odd sum or sum greater than 9) =(18 / 36) + (6/36) = 24/36 = 2/3

P(odd sum and sum greater than 9) = (sample spaces that have an odd sum / total number of sample spaces)  x (sample spaces that have a sum greater than 9 / total number of sample spaces)

P(odd sum and sum greater than 9) = 18/36 x 6/36 = 1/12

P(A) = (sample spaces that have a 3 first / total number of sample spaces) +  (sample spaces that have a 3 first / total number of sample spaces)

P(A) = 6/36 + 6/36 = 1/3

P(B) = (sample spaces that adds up to 7 / total number of sample spaces)

= 6/36 = 1/6

P(A and B) = 1/3 x 1/6 = 1/18

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To fill the pool 3/4 full it will take 240 minutes

A rate is a ratio that compares two different quantities which have different units. For example, if we say John types 50 words in a minute, then his rate of typing is 50 words per minute. The word "per" gives a clue that we are dealing with a rate. The word "per" can be further replaced by the symbol "/" in mathematical problems.

3/4 of the volume the pool is 3/4×16000= 12000gallons

it takes 1 minutes to fill 50gallons

therefore to fill 12000 gallons it will take 12000×1/50 =240minutes

therefore the number of minutes it will take to fill 3/4 of the pool is 240minutes

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Answer:

3

Step-by-step explanation:

19.861>19.851

look at hundreds column

Answer:

60

Step-by-step explanation:

[tex] \sqrt{3} cosecx - 2 = 0[/tex]

[tex] \sqrt{3} cosecx = 2[/tex]

[tex]cosec = \frac{2}{ \sqrt{3} } [/tex]

[tex]sinx = \frac{ \sqrt{3} }{2} [/tex]

[tex]x = sin {?}^{ - 1} ( \frac{ \sqrt{3} }{2} )[/tex]

[tex]x = 60[/tex]

Step-by-step explanation:

Let [tex]{ \green{ \tt{y = \sqrt{ \frac{(x - 3)( {x}^{2} + 4)}{3 {x}^{2} + 4x + 5}}}}} [/tex]

Take log on both sides

[tex]{ \purple{ \sf{ log_{e}y = log_{e} [\frac{(x - 3)( {x}^{2} + 4}{3 {x}^{2} + 4x + 5}]^{ \frac{1}{2}}}} } { \to}{ \tt{ {eq}^{n} (1)}}[/tex]

This above expression is in the form of

[tex]{ \boxed{ \red {\sf{ {x}^{n} = nx}}}}[/tex]

[tex]{ \boxed{ \red{ \sf{ log( \frac{m}{n} ) = log \: m \: - log \: n}}}}[/tex]

Let's apply these two formulas to eqⁿ (1) then,

[tex]{ \purple{ \sf{ log_{e}y = \frac{1}{2}[log(x - 3) + log( {x}^{2} + 4) - log( {3x}^{2} + 4x + 5) ]}}}[/tex]

differentiate with respect to x.

[tex]{ \blue{ \sf{ \frac{1}{y} \frac{dy}{dx} = \frac{1}{2}[ \frac{1}{x - 3} (1 - 0) + \frac{1}{ {x}^{2} + 4 } (2x - 0) - \frac{1}{3 {x}^{2} - 4x + 5} (6x + 4 + 0)]}}}[/tex]

[tex]{ \blue{ \sf{ \frac{dy}{dx} = y[ \frac{1}{2(x - 3)} + \frac{x}{ {x}^{2} + 4} - \frac{3x + 2}{ {3x}^{2} + 4x + 5 } ]}}}[/tex]

where [tex]{ \boxed{ \purple{ \sf{y = \sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{3 {x}^{2} + 4x + 5}}}}} } [/tex]

Answer:

[tex]let \: \: y = \sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } } \\ \\ y =( {{\frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } }})^{ \frac{1}{2} } \\ \\ taking \: \: logarathim \ \: both \: \: sides \: \: \\ \: we \: \:get ⇒ \: ln(y) = {\frac{1}{2} ln(\frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } }) \\ \\ ⇒2 ln(y) = ln(x - 3) + ln( {x}^{2} + 4) - ln(3 {x}^{2} ) + ln(4x) + ln(5) \\ differeciate \: \: both \: \: the \: \: sides \: \: we \: \: get \: \: \\ ⇒ 2 \times \frac{1}{y} (\frac{dy}{dx} ) = \frac{1}{x - 3} + \frac{1}{ {x}^{2} + 4} .2x - \frac{1}{3 {x}^{2} } .6x + \frac{1}{4x} .4 + 0 \\ \\ ⇒ \frac{2}{y} \times \frac{dy}{dx} = \frac{1}{x - 3} + \frac{2x}{ {x}^{2} + 4} - \frac{2}{x} + \frac{1}{x} \\ \\ ⇒ \frac{dy}{dx} = \frac{y}{2} . (\frac{1}{x - 3} + \frac{2x}{ {x }^{2} + 4} - \frac{1}{x} ) \\ \\ put \: \: the \: \: value \: \: of \: \: y \: \: we \: \: get \: \\ ⇒ \frac{dy}{dx} = \frac{\sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } } \\ }{2} \times (\frac{1}{x - 3} + \frac{2x}{ {x }^{2} + 4} - \frac{1}{x} )[/tex]

Step-by-step explanation:

used formula

[tex] { ln(m) }^{n} = n \: ln \: m \\ \\ ln(mn) = ln(m) + ln(n) \\ \\ ln( \frac{m}{n} ) = ln(m) - ln(n) \\ \\ \frac{d}{dx} ( ln(x) ) = \frac{1}{x} [/tex]

The probabilities are given as follows:

a. P(E or F) if E and F are disjoint: 0.9 = 90%.

b. P(E or F) if E and F are independent: 0.7 = 70%.

In a Venn probability, two events are related with each other, as are their probabilities.

The "or probability" is given by:

[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/tex]

Disjoint events are events that cannot happen at the same time, hence P(E and F) = 0, and then the probability is:

P(E or F) = P(E) + P(F) = 0.4 + 0.5 = 0.9 = 90%.

When E and F are independent, we have that:

P(E and F) = P(E)P(F) = 0.4 x 0.5 = 0.2.

Hence the probability in item b is given by:

P(E or F) = P(E) + P(F) - P(E and F) = 0.4 + 0.5 - 0.2 = 0.7 = 70%.

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The function f(x) in standard form as described in the task content is; 3x³ + 21x² - 23x + 5.

It follows from the task content that the function, f(x) which is as described is to be determined.

Since the polynomial remainder theorem postulates that;

Polynomial = (quotient × divisor) + remainder.

According to the given task content;

The quotient is; 3x² - 3x + 1.

The divisor is; (x + 8) and the remainder is; -3.

Therefore, the required polynomial is;

f(x) = (3x² - 3x + 1)(x + 8) + (-3)

= 3x³ + 24x² -3x² -24x + x + 8 - 3

= 3x³ + 21x² - 23x + 5.

Therefore, the required function, f(x) is; 3x³ + 21x² - 23x + 5.

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The given sets written in interval notation are

C = (4, ∞), and  = (-∞, 9]

From the question, we are to write the given sets using the interval notation

The given sets are

C = {v | v > 4}

D = {v | v ≤ 9}

For set C,

C = {v | v > 4}

The elements of the sets are 5, 6, 7, 8, 9, ...

This can be written in interval notation as (4, ∞)

For set D,

D = {v | v ≤ 9}

The elements of the sets are 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, ...

This can be written in interval notation as (-∞, 9]

Hence, the interval notation form of the sets are

C = (4, ∞)

and

D = (-∞, 9]

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Answer:

102.375

Step-by-step explanation:

hope it helps :)

multiply 6.5 x 2.25, get 14.625

multiply 14.625 x 7, answer is 102.375

The company's start-up cost is equal to $1/2530.

The company's minimum cost is equal to $15/37,949.

The number of gadgets that this company must produce and sell in order to incur the least cost is equal to 1/30 gadgets.

From the information provided, the cost C (in dollars) for a company to produce and sell x thousand gadgets is modeled by the following function:

Function, C = 1/(30x² − 2x + 2530)

Mathematically, the start-up cost for this company is equal to the total cost when no sales of its gadget has been made i.e the level of activity is at zero (x = 0). Therefore, we would substitute the value of x in the function with zero (0):

Start-up cost = 1/(30(0)² − 2(0) + 2530)

Start-up cost = $1/2530.

First of all, we would determine the number of gadgets that this company must produce and sell in order to incur the least cost by differentiating the function for the total cost using the quotient rule:

dC/dx = [(0(30x² − 2x + 2530) - 1(60x - 2))1/(30x² − 2x + 2530)²]

dC/dx = [(0 - 60x + 2)1/(30x² − 2x + 2530)²]

Equating the marginal cost function to zero, we have:

-60x + 2/(30x² − 2x + 2530)² = 0

-60x + 2 = 0

60x = 2

x = 2/60

x = 1/30 gadgets.

Now, we can determine the minimum cost:

Function, C = 1/(30x² − 2x + 2530)

Minimum cost = 1/(30(1/30)² − 2(1/30) + 2530)

Minimum cost = 1/(1/30 - 1/15 + 2530)

Minimum cost = 1/(-1/15 + 2530)

Minimum cost = 1/(37,949/15)

Minimum cost = $15/37,949

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Answer:

[tex]y = \frac{1}{4} x\\[/tex] [tex]- 5[/tex]

Step-by-step explanation:

Take a look at the attachment...

Hope this helps! :)

Step-by-step explanation:

(a)

it just means how many members are in a set.

there is no mystery or strange mathematical approach. we simply need to count them.

when there is an interval given, sure, we can calculate the length. but for a short interval counting is still faster.

n(A) = 6

n(B) = 3 {-1, 0, 1}

the calculation would be 2 - -2 - 1 = 4 - 1 = 3.

the "-1" comes from the fact that the end of the interval is not included.

(b)

z is not in A. as we can see : true

l is in A. as we can see again : yes. true.

-1 is in B. as I listed above : yes, true.

7 is in B. no, definitely not. false.

Answer: 48 qt

Step-by-step explanation:

The conversion factor between pints and quarts is two which means there are 2 pints in every quart. divide the number of pints by two to find the number of quarts.

96/2 = 48

EX: 24 pints = 12 quarts

24/2 =12

EX 8 quarts = 16 pints

The same rule works in reverse but instead of dividing you multiply.

8x2 = 16

Polynomial of degree 4 in factored form is x(x+3)(x-4)(x-8) = 0

The Standard form of polynomial with degree 4 is ax⁴ + bx³ + cx² + dx + e

Zeros of the polynomial is the points where the polynomial becomes zero.

According to the question,

The zeros of the polynomial are -3 , 4 , 0 , 8

Let equation of polynomial f(x) be

f(x) = ax⁴ + bx³ + cx² + dx + e ------------------(1)

Sum of zeros = -b/a = -3 + 4 + 0 + 8

= 9

Sum of product of two zeros = c/a =  (-3× 4 ) +(0 × 4) + (0 × 8) + (8× -3)

= -12 + 0 + 0 -24

= -12 - 24

= -36

Sum of product of sum of three zeros = -d/a = ( -3×4×0) + ( 4×0 ×8 ) + (0×8×-3) + (-3×4×8)

= 0 - 96

= -96

Product of all four zeros = e/a = (-3×4×0×8)

= 0

So,  b = -9 , c =-36 , d = 96 and e = 0

Substituting value in equation (1),

Hence , the required polynomial = x⁴ -9x³ -36x²+96x

Factored form ,

x⁴ -9x³ -36x² +96x = 0

=> x(x³ - 9x² - 36x + 96) = 0

=> x(x+3)(x-4)(x-8) = 0

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The range of this quadratic function y = x² - 2x - 3 is all real numbers greater than 2.

Suppose we have an ordered pair (x, y) then the domain of the function is the set of values of x and the range is the set of values of y for which x  is defined.

We know a quadratic function y = ax² + bx + c is a graph of a parabola that opens upwards as the coefficient of x² is positive.

Given,  y = x² - 2x - 3.

y = x² - 3x + x - 3.

y = x(x - 3) + 1(x - 3).

y = (x - 3)(x + 1).

Now the x-intercepts are at 3 and -1 so the vertex is located at (3 - 1)/2 = 2.

So the range of this function is all real numbers greater than 2 as the parabola opens upwards.

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The ordering and transportation cost C for the components used in manufacturing a product is c= 100(200/x^2+ x/x+30), x≥1 where C is measured in thousands of dollars and x is the order size in hundreds.

To find the rate of change of C with respect to x when (a) x = 10, (b) x = 15, and (c) x = 20.

The rate of change of the cost with respect to x is its derivative.

The differentiation rules to apply are

Quotient Rule: [tex]\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f^{'}(x)-f(x)g^{'}(x)}{\left[g(x)]^{2} }[/tex]

Constant Multiple Rule: [tex]\frac{d}{dx} \left[cf(x) \right] = cf^{'}(x)[/tex]

Power Rule: [tex]\frac{d}{dx} \left[x^{n} \right] = nx^{n-1}[/tex]

Constant Rule: [tex]\frac{d}{dx} [c] = 0[/tex]

So,

[tex]c^{'}(x) = 100 (\frac{x^{2}(200)^{'}-200(x^{2})^{'}}{(x^{2})^{2} }+\frac{(x+30)(x)^{'}-x(x+30)^{'} }{(x+30)^{2} } )[/tex]

[tex]c^{'}(x) = 100 (\frac{x^{2}(0)-200(2x) }{x^{4} } + \frac{(x+30)(1)-x(1+0)}{(x+30)^{2} })[/tex]

[tex]c^{'}(x) = 100 (\frac{0-400x}{x^{4} } + \frac{x+30-x}{(x+30)^{2} })[/tex]

[tex]c^{'}(x) = 100 (\frac{-400x}{x^{4} } + \frac{30}{(x+30)^{2} })[/tex]

From the above values it appears that the rate of change of the cost increases as the size of the order increases

[tex]a) c^{'}(10) = -38.13\\ \\b) c^{'}(15) = -10.37\\ \\c) c^{'}(20) = -3.8[/tex]

Hence the above values it appears that the rate of change of the cost increases as the size of the order increases

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1,858.40

2300-36%=1472

1472-70%=441.60

2300-441.6=1,858.40

Answer:

∠ 4 = 160° , ∠ 6 = 20°

Step-by-step explanation:

∠ 4 and ∠ 6 are same side interior angles and sum to 180°

given ∠ 6 = [tex]\frac{1}{8}[/tex] ∠ 4 ( multiply both sides by 8 to clear the fraction )

8 ∠ 6 = ∠ 4

then

∠ 4 + ∠ 6 = 180° , that is

8 ∠ 6 + ∠ 6 = 180°

9 ∠ 6 = 180° ( divide both sides by 9 )

∠ 6 = 20°

∠ 4 = 8 × ∠ 6 = 8 × 20° = 160°

The equation of the tangent line of the function is y = x + 20

f(x)=(x^2+6)(x−4)

expanding the function gives

f(x)=x^3 - 4x^2 + 6x - 24

At a specific point the slope, m is the derivative

finding the derivative of f(x)

f'(x) = 3x^2 - 8x + 6

The derivative at the point (1, -21) is found by substituting the point to the equation

f'(1) = 3 * 1 - 8 * 1 + 6 = 1

the slope, m = 1

equation of the line of slope = 1 passing through point (1, -21) is calculated using the point slope formula which is

(y - y') = m(x - x')

(y - -21) = 1(x - 1)

y + 21 = x - 1

y = x + 20

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1) The amount of time that it will take to have 50 stamps is; 10 months

2) The amount of money that bella will have after 12 months is; $112

1) We are told that Jack received a collection of 30 baseball cards. Now, he decided to start collecting after that and he adds 2 stamps every month. Thus, let m represent the number of months he collects and so we have the equation as;

2m + 30

Now, we want to find out how long it will take to get 50 stamps. Thus;

2m + 30 = 50

2m = 20

m = 10 months

2) Principal = $100

Interest rate = 1% monthly

Thus;

Interest after 12 months = 100(0.01 * 12) = $12

Thus, amount she now has after 12 months = $100 + $12 = $112

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Answer:

C

Step-by-step explanation:

blue volume = 10

red volume = 6

total = 10  6 = 16

ratio of

blue : total

= 10 : 16 ( divide both parts by 2 )

= 5 : 8

A) The monthly deposit is $503.77

B) Interest per month ≈0.42

what is compound interest?

Interest that is added to a loan or deposit sum is known as compound interest. In our daily lives, it is the notion that is employed the most frequently. Compound interest is calculated for a sum using the principal and interest accrued over time. Compound interest and simple interest differ primarily in this way.

Let A be the amount in future.

Monthly deposit= [tex]P= \frac{A \times \frac{r}{n}} {(1 +\frac{r}{n})^{t*n} - 1}[/tex]

n= 12, 12 months in a year

So, monthly deposit= [tex]P= \frac{300000 \times \frac{0.05}{12}} {(1 +\frac{0.05}{12})^{25*12} - 1}[/tex]

                                 = 1250/2.4813

                                 = $503.77

So, monthly deposit= $503.77

To find interest:

Interest per year = 5%

Interest per month = 5/12 = 0.416 ≈0.42

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Answer: $503.77

Interest earned: $148,869

Step-by-step explanation:

Let A be the future amount.

Monthly Deposit= P=A*r/n÷ (1+r/n)^t*n-1

n=12 months in a year

P= (300,000*0.05/12)÷ [(1+0.05/12)^25*12]-1

P=1250/2.4813

P=$ 503.77

Interest earned.

$300,000-(503.77*12*25)

$300,000-151,131= $148,869

Number of countries would west Virginia be expected to have is  18 countries

The number of campgrounds that North Dakota has = 465 campgrounds

The number of countries = 12 countries

The number of campgrounds that West Virginia has = 711 campgrounds

Given that the North Dakota was proportional to West Virginia in the number of campgrounds to counties.

The proportion will be

465 / 12 = 711 / x

Divide the terms

38.75 = 711 / x

x = 711 / 38.75

Divide the terms

x = 18.4

x ≈ 18 countries

Hence, number of countries would west Virginia be expected to have is  18 countries

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Answer:

Step-by-step explanation:

C E F or just E and F

Answer:

y = -9

Step-by-step explanation:

The slope-intercept form is y = mx + b

From two points we can easily determine the slope, m = (y2-y1)/(x2-x1)

You would put in the values into the equation.

m = (-9-(-9))/(-3-1) = 0/-4 = 0

Using the slope-intercept form, y = 0(x) + b ⇒ y = b

But we don't know what the y-intercept (b) is yet.  Using the point-slope form we can determine b.

Point-slope: (y - y1) = m(x - x1)

(y - (-9)) = 0(x - (-3))

y + 9 = 0

y = -9

So the equation is y = -9.