Karen can read 10 pages and complete 15 math problems in 30 minutes. How long will it take her to do her homework if she has 30 pages to read and 45 math problems to complete? Your answer should refer to the amount of time in minutes.

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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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]

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

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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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.

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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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 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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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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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.

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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Answer:

HF ≈ 71.4

Step-by-step explanation:

since the triangles are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{HF}{EC}[/tex] = [tex]\frac{FG}{CD}[/tex] ( substitute values )

[tex]\frac{HF}{23}[/tex] = [tex]\frac{59}{19}[/tex] ( cross- multiply )

19 HF = 1357 ( divide both sides by 19 )

HF ≈ 71.4 ( to the nearest tenth )

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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1,858.40

2300-36%=1472

1472-70%=441.60

2300-441.6=1,858.40

Answer:

[tex]s_(_t_)=\frac{1}{3}t^3+\frac{7}{2}t^2-8t+5[/tex]

Step-by-step explanation:

1.)

[tex]a_t=v'_t\\v_t=\int(a_(_t_)dt\\v_t=\int(2t+7)dt\\v_t=t^2+7t+C\\[/tex]

2.)

[tex]v_0=-8\\-8=(0)^2+7(0)+C\\C=-8\\v_t=t^2+7t-8[/tex]

3.)

[tex]v_t=s'_t\\s_t=\int(v_t)dt\\s_t=\int(t^2+7t-8)dt\\s_t=\frac{1}{3}t^3+\frac{7}{2}t^2-8t+C\\[/tex]

4.)

[tex]s_0=5\\5=\frac{1}{3}(0)^3+\frac{7}{2}(0)^2-8(0)+C\\C=5\\s_t=\frac{1}{3}t^3+\frac{7}{2}t^2-8t+5[/tex]

The meaning of the symbol 'n,' represents the total number of individuals or observations in the sample.

Statistics is the science that studies and develops methods for collecting, analyzing, interpreting, and presenting empirical data. Statistics is the study and manipulation of data, which includes methods for gathering, reviewing,

analyzing, and drawing conclusions from data. Statistics is divided into two categories: descriptive statistics and inferential statistics.

The primary goal of using statistics is to plan the data collected in terms of experimental designs and statistical surveys. Statistics is a mathematical science concerned with numerical data. In short, statistics is an important process that assists in making data-driven decisions.

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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:

∠ 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°

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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Answer:

Step-by-step explanation:

C E F or just E and F

Answer:

The rocket reaches its maximum height at 2.25 second(s) after launch.

The maximum height reached by the object is 88 feet.

Step-by-step explanation:

How long will it take the rocket to reach its maximum height?

h(t)=-16t² + 72t +7 is a quadratic equation in the form at²+bt+c, where a=-16 and b=72 and c=7.

The maximum height would be at the vertex / peak of this equation, and we can find the x-coordinate (aka the value of 't') at this point using the formula [tex]\frac{-b}{2a}[/tex]:

[tex]t=\frac{-b}{2a} =\frac{-72}{2(-16)} =\frac{72}{32} =2.25seconds[/tex]

What is the maximum height?

Since we know how long it takes to reach maximum height, we can sub this value into the equation to find our answer:

h(t)=-16t² + 72t +7

h(2.25)=-16(2.25)² + 72(2.25) +7

=-81+162+7=88feet

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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2 1/2 cups of milk does she need to make 6 batches of waffle mix.

Given:

A total of 12 waffles can be made from each batch of waffle mix.

Cynthia is making 6 batches for a breakfast board meeting. If the recipe requires 5/12 cups of milk for each batch.

Number of cups of milk required = number of batches*milk required for each batch.

= 6 * 5/12

= 30/12

= 15/6

= 5/2

= 2 1/2 cups.

Therefore 2 1/2 cups of milk does she need to make 6 batches of waffle mix.

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Answer: 88 grams

Step-by-step explanation:

The total weight of the 30 apples is [tex](30)(99)=2970[/tex] grams.

The total weight of all the apples in the box and bag is [tex]55(94)=5170[/tex] grams.

So, the weight of the apples in the bag is [tex]5170-2970=2200[/tex] grams.

Thus, the mean weight is [tex]\frac{2200}{25}=88[/tex] grams.

If 3/8 fraction of the dishes were from China and another 12.5 percent from Japan, it implies that 50 percent were from other countries.

Firstly, we convert the proper fraction to a percentage by expressing it over 100.

Secondly, the percentages of China and Japanese dishes are added before being subtracted from 100%.

The resulting difference is the percentage of dishes from other countries.

In summary, unit conversion, expressing the fraction as a percentage, and mathematical operations of addition and subtraction are involved.

Dishes from China = 3/8 = 0.375 or 37.5% (3/8 x 100)

Dishes from Japan = 12.5%

Dishes from other countries = 50% (1 - 37.5% - 12.5%)

Thus, the percentage of dishes from other countries is 50%.

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At a food festival, 3/8 of the dishes were from China. Another 12.5% of the dishes were from Japan.

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

The discounted price of that jeans on Valentine's day is Php 780.

From the information, the original sale price of Levi's jeans at Sm Department store is Php 1 300.00 and on February 14, all jeans will have a 40% discount.

The amount of the discount will be:

= Percentage of discount × Sales amount

= 40% × 1300

= 0.4 × 1300

= 520

The discount is Php 520

Discounted Price = 1300 - 520

= Php 780

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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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