Fill in the table.
X(input) Y(output)
1 0
3 4
4 6
7
20

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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For each additional slice of pizza that is purchased, the total cost of lunch is increase by $2.5

First choose two points

(1, 10) and (2, 12.5)

The cost of one slice of pizza = $10

The cost of two slice of pizza = $12.5

Using these points we have to find the slope of the line

The slope of the line is the change in y coordinates with respect to the change in x coordinates of the graph

The slope of the line m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Where m is the slope of the line

[tex](x_1,y_1)[/tex] is the coordinates of the first point

[tex](x_2,y_2)[/tex] is the coordinates of the second point

Substitute the values in the equation

Slope of the line = (12.5 - 10) / (2-1)

= 2.5/1

= $2.5

Hence, for each additional slice of pizza that is purchased, the total cost of lunch is increase by $2.5

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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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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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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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Answer: 573,948 memberships

Step-by-step explanation:

1. Multiply the number of memberships the year prior by 80%

318,860 * 0.8 = 255088

If they sold 80% more memberships this year, that means you would add 80% of the previous year's memberships (255088) to the number of memberships sold last year (318860).

2. Add 80% of the previous year's memberships to the number of memberships sold last year

255088 + 318860 = 573948

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.

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 )

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 value of x and y will be x =  16° and y = 20°

What is Angles?

In Euclidean geometry, an angle is the shape created by two rays that share a terminus and are referred to as the angle's sides and vertices, respectively. The plane that contains the rays contains the angles created by two beams. The meeting of two planes might also result in an angle. Dihedral angles are what these are.

We can write

(6x + y) = (x + 5y)

[By corresponding angles]

6x + y = x + 5y

6x - x = 5y - y

5x = 4y

x = 4y/5

Now by linear pair

4x + (6x + y) = 180

4x + 6x +y = 180

10x + y = 180

By putting the value of x

10(4y/5) + y = 180

8y + y = 180

9y = 180

y = 180/9

y = 20°

Putting the value of y in x we get

x = 4(20) / 5

x = 4(4)

x = 16°

Hence, value of x =  16° and y = 20°

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Answer:

Step-by-step explanation:

C E F or just E and F

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°

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.

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:

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 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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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 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:

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 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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Answer:

x = -1

Step-by-step explanation:

The average number of liters per 100 km used by Petra's car during the trip, using proportions, was of:

13.96 liters per 100 km.

A proportion represents a fraction relative to a total amount, and this fraction, which has the meaning of a unit rate, is used along basic arithmetic operations, especially multiplication and division, to find the desired measures in the context of a problem.

The total amount of liters used by Petra on her trip was of:

45 + 35.45 + 30.5 + 30 = 140.95 liters.

The last palindromic number in the twenty-two thousands is of:

22922.

Hence the total distance traveled by Petra was of:

22,922 - 21,912 = 1010 km.

The rate of liters per km is of:

140.95/1010 = 0.1396.

Hence the rate of liters per 100 km is:

0.1396 x 100 = 13.96 liters per 100 km.

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The area of the trapezoid is 26 square units

Area of trapezoid

= (sum of parallel lines) /2 * height

Length of the longer parallel line is solved using Pythagoras theorem is applicable to right angle triangle.  the formula of the theorem is

hypotenuse² = opposite² + adjacent²

hypotenuse = 5

opposite = 4

adjacent = ?

hypotenuse = 7.2

opposite = 4

adjacent = ?

adjacent = √(hypotenuse² - opposite²  )

adjacent 1 = √(5² - 4² ) = 3

adjacent 2 = √(7.2² - 4² ) = 5.987 ≅ 6

length of the longer parallel line

= 3 + 2 + 6

= 11

Area of trapezoid

= (sum of parallel lines) /2 * height

= (2 + 11) /2 * 4

= 6.5 * 4

= 26 square units

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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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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:

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 procedure to calculate the p-value from z-score depends on one of two choices: one-tailed (which may be Left-tailed or Right-tailed) or two-tailed tests.

A z-table, otherwise termed the standard normal table, provides the area under the curve to the left of a z-score.

The technique for using z-tables to find p-values depends on the test being carried out; which may be; one- or two-tailed test.

For one-tailed tests, the directionality is also a factor.

It therefore follows that;

For a Left-tailed z-test: the p-value = Φ(Zscore)

For a Right-tailed z-test: the p-value = 1 - Φ(z-score).

For a Two-tailed z-test: the p-value = 2 * Φ(−|z-score|).

The procedure described above represents how to calculate p-value from z-score.

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The following equations are used to find the p-value from its z-score

From the question, the parameters are given as

Also, from the question, we understand that:

The p value depends on the z-score for its own value

There are three formulas for calculating the p-value from the z-score, depending on the type of test (i.e. left-tailed test, right-tailed test and two-talied test)

These formulas are:

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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]