Among all pairs of numbers with a sum of 232, find the pair whose product is maximum. Write your answers as fractions reduced to lowest terms.

Question:

At the city museum, child admission is $5.80 and adult admission is $9.30. On Monday, four times as many adult tickets as child tickets were sold, for a total sales of $1548.00. How many child tickets were sold that day?

Answer:

36 child tickets were sold

Solution:

Given that,

Cost of 1 child admission = $ 5.80

Cost of 1 adult admission = $ 9.30

Let "c" be the number of child tickets sold

Let "a" be the number of adult tickets sold

On Monday, four times as many adult tickets as child tickets were sold

Number of adult tickets sold = four times the number of child tickets

Number of adult tickets sold = 4(number of child tickets sold)

a = 4c ----- eq 1

They were sold for a total sales of $ 1548.00

number of child tickets sold x Cost of 1 child admission + number of adult tickets sold x Cost of 1 adult admission = 1548.00

[tex]c \times 5.80 + a \times 9.30 = 1548[/tex]

5.8c + 9.3a = 1548  ---- eqn 2

Let us solve eqn 1 and eqn 2 to find values of "c" and "a"

Substitute eqn 1 in eqn 2

5.8c + 9.3(4c) = 1548

5.8c + 37.2c = 1548

43c = 1548

c = 36

Thus 36 child tickets were sold that day

Answer:

$8,100

Step-by-step explanation:

Tyler owns a major medical policy with 70/30 coinsurance and a $3,000 deductible. If he submits a claim for $20,000, how much will he pay?

Tyler pays $8,100.

It can be calculated thus as $20,000 - $3,000 = $17,000.

So $17,000 x .30 = $5,100.

The sum of  $3,000 deductible with the $5,100 coinsurance

($3,000 + $5,100= $8,100).

The correct answer is: $8,100  

Insurance is a from of protection against financial loss. it is a type of risk management strategy used by businesses and individual entities.The entity that provides insurance is known as an Insurer

Answer:

B

Step-by-step explanation:

4x-5y=2 ...(1)

10x-21y=10   ...(2)

(1)+2(2)  gives

24x-47y=22

The equivalent system for the given system of equations is 4x − 5y = 2, and 10x − 21y = 10 is 4x − 5y = 2  and 24x − 47y = 22

We have to determine, the equivalent system for the given system of equations is 4x − 5y = 2, and 10x − 21y = 10.

According to the question,

System of equation; 4x − 5y = 2, and 10x − 21y = 10.

To determine the equivalent relation following all the steps given below.

From equation 1,

[tex]4x - 5y = 2\\\\4x = 2 +5y\\\\x = \dfrac{2+5y}{4}[/tex]

Substitute the value of x in equation 2,

[tex]10(\dfrac{2+5y}{4} )- 21y= 10\\\\10 + 25y - 42y = 10 \times 2\\\\10 -17y = 20\\\\-17y = 20-10\\\\-17y = 10\\\\ y = \dfrac{-10}{17}[/tex]

Substitute the value of y in equation 1,

[tex]4x - 5(\dfrac{-10}{17}) = 2\\\\68x + 50 = 34\\\\68x = 34-50\\\\68x = -16\\\\x = \dfrac{-16}{68}\\\\x = \dfrac{-4}{17}[/tex]

The equation is equivalent to given relation which satisfies the value of x and y is,

[tex]-47(\dfrac{-10}{17} )+24 (\dfrac{-4}{17})= 22\\\\\dfrac{470-96}{17} = 22\\\\\dfrac{374}{17} = 22\\\\22 = 22[/tex]

Hence, The equivalent system for the given system of equations is 4x − 5y = 2, and 10x − 21y = 10 is 4x − 5y = 2  and 24x − 47y = 22.

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Divide the first number by the first number in the ratio, then the second number and see which ones are the same:

1.  

24/3 = 8, 54/9 = 6

18/3 = 6, 54/9 = 6

36/3 = 12, 81/9 = 9

The answer is 18:54

2. 18/30 = 0.6

The ratio needs to be equal to 0.6

2/3 = 0.66

3/5 = 0.6

4/5 = 0.8

The answer is 3:5

3. Divide Y by X:

24/18 = 1.33

The Y value is the X value multiplied by 1.33

Y = 48 x 1.333 = 64

The answer is 64

Answer:

Step-by-step explanation:

Let L represent the length of the rectangular porch.

Let W represent the width of the rectangular porch.

The area of the rectangular porch is expressed as LW.

Bills new porch is rectangular with an area of 50 square feet. Therefore,

LW = 50 - - - - - - - - 1

if the length is two times the width, it means that

L = 2W

Substituting L = 2W into equation 1, it becomes

2W × W = 50

2W^2 = 50

W^2 = 50/2 = 25

W = √25 = 5

LW = 50

5L = 50

L = 50/5 = 10

The perimeter of he rectangle is

Perimeter = 2(L + W)

Perimeter = 2(10 + 5) = 2 × 15

Perimeter = 30 feet

The value of a²+b² = -99/2.

Add the given equations:

a³ - 3ab² + b³ - 3a²b = 47 + 52

(a³ + b³) - 3ab(a + b) = 99

Factor the sum of cubes:

(a + b)(a² - ab + b²) - 3ab(a + b) = 99

(a + b)(a² - 4ab + b²) = 99

Square both given equations:

a⁶ - 6a⁴b² + 9a²b⁴ = 47²

b⁶ - 6a²b⁴ + 9a⁴b² = 52²

Add these two squared equations:

a⁶ + b⁶ - 6a²b⁴ + 9a²b⁴ - 6a⁴b² + 9a⁴b² = 47² + 52²

a⁶ + b⁶ + 3a⁴b² + 3a²b⁴ = 47² + 52²

Factor using sum of cubes:

(a² + b²)³ = 47² + 52²

Take the cube root of both sides:

a² + b² = ³√(47² + 52²)

Evaluate the cube root:

a² + b² ≈ -99/2

Answer:

2 fluid ounce/hour

Step-by-step explanation:

2 hours to 5 hours ; 3 hours apart

leaks: 48 -42 = 6 fluid ounce

rate of leak = 6/3 = 2 fluid ounce/hour

The rate at which water is leaking from the jug will be 2 ounces per hour.

It is the average amount by which the function is modified per unit throughout that time period. It is calculated using the gradient of the line linking the interval's ends on the graph that represents the function. The average rate of change of the function is given as,

Average rate = [f(x₂) - f(x₁)] / [x₂ - x₁]

Water is spilling from a container at a steady rate. Subsequent to spilling for 2 hours, the container contains 48 liquid ounces of water. Subsequent to spilling for 5 hours, the container contains 42 liquid ounces of water.

Then the rate at which water is leaking from the jug will be given as,

Rate = |(42 - 48) / (5 - 2)|

Rate = |-6 / 3|

Rate = 2 ounces per hour

The rate at which water is leaking from the jug will be 2 ounces per hour.

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Answer: C. framing

Step-by-step explanation:

People tends to decides on options based on the type of framing presented to them. Framing effect is a cognitive bias where people decide on options presented to them based on whether it's presented with positive or negative connotations and remarks. In the case above, the reaction of people to the same idea when presented positively and negatively was different. It implies that the framing of the same idea may influence people's decision

Answer:

When you reflect a point across the line y = x, the x-coordinate and y-coordinate change places. If you reflect over the line y = -x, the x-coordinate and y-coordinate change places and are negated (the signs are changed). the line y = x is the point (y, x). the line y = -x is the point (-y, -x).

Answer:

Slope intercept form -   y = −2x

Slope is m = −2  

Slope

Answer:

C 1 8/12 + 2 3/12 + 2/12 =

Step-by-step explanation:

Constituents of the fruit salad prepared by Emily:

This can be expressed as follows:

[tex]\[1\frac{2}{3}+2\frac{1}{4}+\frac{1}{6}\][/tex]

This can be equivalently expressed as :

[tex]\[1\frac{8}{12}+2\frac{3}{12}+\frac{2}{12}\][/tex]

Among the given options, this corresponds to option C.

Answer:

Maximum 3 numbers. Minimum 1 number.

Step-by-step explanation:

Well, let us look at the case when 3 numbers are greater than 30. Let us take numbers as 1, 2, 3 and 114 and find their mean which is (1+2+3+114)/4=30.

Now let us look at the case in which 2 numbers greater than 30. Let us take numbers 28, 29, 31 and 32 and find their mean which is (28+29+31+32)/4=30.

Now let us look at the case in which 1 number greater than 30. Let us take numbers 27, 28, 29 and 36 and find their mean which is (27+28+29+36)/4=30.

So it can be concluded that maximum 3 numbers and minimum 1 number are greater than 30.

Answer:

Step-by-step explanation:

Let x represent the number of hours that Justin works in a week.

Let y represent the total amount that Justin would receive for working for x hours.

Justin earns $8 an hour for the first 40 hours he works and $12 for each additional hour. This means that the total amount that he earns in a week would be

y = 8×40 + 12(x - 40)

y = 320 + 12(x - 40)

If he earns 48 hours in a week, the total amount that he earned would be

320 + 12(48 - 40) = $416

Answer:

$50

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let p represent the original price of the computer game.

Let s represent the sales price of the computer game.

Jacob found a computer game that was on sale at 20% off its original price. This means that the amount that was taken off the original price would be

20/100 × p = 0.2 × p = 0.2p

The expression for the sale price would be

s = p - 0.2p

s = 0.8p

Answer:

Step-by-step explanation:

The equation of a straight line can be represented in the slope intercept form as

y = mx + c

Where

m = slope = (change in the value of y in the y axis) / (change in the value of x in the x axis)

The equation of the given line is

7x+4y=3

4y = - 7x + 3

y = -7x/4 + 3/4

Comparing with the slope intercept form, slope = -7/4

If the line passing through the given point is perpendicular to the given line, it means that its slope is the negative reciprocal of the slope of the given line.

Therefore, the slope of the line passing through (6,-9) is 4/7

To determine the intercept, we would substitute m = 4/7, x = 6 and y = -9 into y = mx + c. It becomes

- 9 = 4/7×6 + c = 24/7 + c

c = - 9 - 24/7 = -87/7

The equation becomes

y = 4x/7 - 87/7

Answer:

Step-by-step explanation:

P

Answer:

Option (C).

The twentieth term of the given arithmetic sequence is -36.

Step-by-step explanation:

The given arithmetic sequence is,

21, 18, 15, 12, ...........

Now, the first term of the arithmetic sequence, a₁ = 21

Second term of the arithmetic sequence, a₂ = 18

Third term of the arithmetic sequence, a₃ = 15

Fourth term of the arithmetic sequence, a₄ = 12

and so on.

Now, common difference, d = a₂ - a₁ = 18 - 21 = -3

We know that, [tex]n^{th}[/tex] term of an arithmetic sequence is given by,

aₙ = a₁ + (n - 1)d

To find the [tex]20^{th}[/tex] term of the given arithmetic sequence, we will substitute the values of a₁ , n and d in the above expression of aₙ.

Put a₁ = 21; n = 20 and d = -3 in the above expression of aₙ, we get

[tex]a_{20}=21+(20-1)(-3)=21+19\times(-3)=21-57=-36[/tex]

So, twentieth term of the given arithmetic sequence is -36.

Hence, option (C) is the correct answer.

Final answer:

The twentieth term of the arithmetic sequence is 78.

Explanation:

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. In this case, the common difference is -3. To find the twentieth term of the sequence 21, 18, 15, 12, ... , we use the formula for the nth term of an arithmetic sequence: an = a1 + (n-1)d

Plugging in the values, we get: a20 = 21 + (20-1)(-3) = 21 + 57 = 78

Therefore, the twentieth term of the arithmetic sequence is 78.

Answer:

K

Step-by-step explanation:

Values of x and y are either negative or positive, but not 0. Lets try to make each choice "negative", so we can eliminate it.

F. x - y

If y is greater than x in any positive number, the result is negative.

1 - 3 = -2

So, this can be negative.

G. |x| - |y|

Here, if y > x for some positive number, we can make it negative. Such as shown below:

|5| - |8|

= 5 - 8

= -3

So, this can be negative.

H.

|xy| - y

Here, if y is quite large, we can make this negative and let x be a fraction. So,

|(0.5)(10)| - 10

|5| - 10

5 - 10

-5

So, this can be negative.

J. |x| + y

This can negative as well if we have a negative value for y and some value for x, such as:

|7| + (-20)

7 - 20

-13

So, this can be negative.

K. |xy|

This cannot be negative because no matter what number you give for x and y and multiply, that result WILL ALWAYS be POSITIVE because of the absolute value around "xy".

So, this cannot be negative.

Final answer:

The expression that cannot be negative for all nonzero values of x and y is K. |xy|. This is because the absolute value of any number, including the product xy, is always nonnegative.

Explanation:

Among the given options, K. |xy| is the expression that cannot be negative for all nonzero values of x and y. The reason for this is that the absolute value of any real number, including the product xy, is always nonnegative. This is due to the definition of absolute value, which measures the magnitude or distance of a number from zero on the number line, disregarding the direction (positive or negative). Therefore, even if x or y or both are negative, resulting in a negative product, the absolute value symbol converts this to a positive value. This fundamental property of absolute values ensures that K. |xy| will always return a nonnegative result, making it impossible to be negative.

Answer:

-96 + 672\,i

Step-by-step explanation:

This is a product of complex numbers, so we have in mind not only the general rules for multiplying binomials, but also the properties associated with the powers of the imaginary unit "i", in particular [tex]i^2=-1[/tex]

We start by making the first product indicated which is that of a pure imaginary number (-6i) times the complex number (8-6i). We use distributive property and obtain the new complex number that results from this product:

[tex]-6\,i\,(8-6\,i)= (-6\,i)\,* 8 \, -\,6\,i\,(-6\,i)=-48\,i+36\,i^2=-48\,i+36\,(-1)=-36-48\,i[/tex]

Now we make the second multiplication indicated (using distributive property as one does with the product of binomials), and combine like terms at the end:

[tex](-36-48\,i)\,(-8-8\.i)=(-36)\.(-8)+(-36)(-8\,i)+(-48\,i)\,(-8)+(-48\,i)(-8\,i)=\\=288+288\,i+384\,i+384\,i^2=288+288\,i+384\,i+384\,(-1)=\\=288-384+288\,i+384\,i=-96+672\,i[/tex]

Answer:

[tex]48.5654 \textdegree[/tex]

Step-by-step explanation:

Let [tex]D[/tex] be the mid point of [tex]AB[/tex]

Now in [tex]\Delta ACD\ and\ \Delta BCD[/tex]

[tex]AC=CB \ (given)\\CD=CD \ (common\ side)\\AD=DB \ (D\ is\ mid\ point\ of\ AB)[/tex]

[tex]Hence\ \Delta ACD\cong\Delta BCD[/tex]

[tex]\angle A=\angle B\\\angle ACD=\angle BCD\\\angle ADB=\angle BDC[/tex]

[tex]\angle ADB+\angle BDC=180\\2\angle ADB=180\\\angle ADB=90[/tex]

[tex]in \Delta BCD\\\cos\angle B=\frac{BD}{BC}\\ =\frac{45}{2\times34}\\ =\frac{45}{68} \\\angle B=\cos^{-1}(\frac{45}{68} )\\\angleB=48.5654\textdegree[/tex]

Answer:

The height of the prism is 7 unit  

Step-by-step explanation:

Given as :

The volume of right triangle prism = v = 21 cubic unit

The length of one base = [tex]b_1[/tex] = 2 unit

The length of other base = [tex]b_2[/tex] = 3 unit

Let The height of the prism = h unit

Now, According to question

Volume of prism = [tex]\dfrac{1}{2}[/tex] ×  [tex]b_1[/tex] ×  [tex]b_2[/tex]× height

Or, v =  [tex]\dfrac{1}{2}[/tex] ×  [tex]b_1[/tex] ×  [tex]b_2[/tex]× h

Or, 21 cubic unit =  [tex]\dfrac{1}{2}[/tex] × 2 unit × 3 unit × h unit

Or, 21 =  [tex]\dfrac{1}{2}[/tex] × 6 × h

Or, 21 = 3 × h

∴   h = [tex]\dfrac{21}{3}[/tex]

i.e h = 7 unit

So,The height of the prism = h = 7 unit

Hence, The height of the prism is 7 unit  Answer

Answer:

The base (b) of the triangle is  

✔ 3

units.

The height (h) of the triangle is  

✔ 5

units.

The area of the triangle is  

✔ 7.5

square units.

Step-by-step explanation:

Answer:

-273

222

217

212

207

202

197

192

187

182

177

172

167

162

157

152

147

142

137

132

127

122

117

112

107

102

97

92

87

82

77

72

67

62

57

52

47

42

37

32

27

22

17

12

7

2

-3

-8

-13

-18

-23

-28

-33

-38

-43

-48

-53

-58

-63

-68

-73

-78

-83

-88

-93

-98

-103

-108

-113

-118

-123

-128

-133

-138

-143

-148

-153

-158

-163

-168

-173

-178

-183

-188

-193

-198

-203

-208

-213

-218

-223

-228

-233

-238

-243

-248

-253

-258

-263

-268

-273

Step-by-step explanation:

Answer:

[tex]u_{n} = a + (n - 1)d\\\\n = 100, a = 222, d = -5\\\\

Substitute the values in.\\\\

u_{100} = 222 + (100 - 1)(-5)\\\\

u_{100} = 222 + (99)(-5)\\\\

u_{100} = 222 + (99)(-5)\\\\

u_{100} = 222 +-495\\\\

u_{100} = -273[/tex]

Answer:

No. If 5x=y+7 then xy=6 and (2) x and y are consecutive integers with the same sign. for xy=6

Step-by-step explanation:

For the sake of clarity:

If 5x=y+7 then (x – y) > 0?

Alternatives:

(1) xy = 6  

(2) x and y are consecutive integers with the same sign

1) Consider (x-y)>0 as true:

[tex]xy=6[/tex] Numbers like, 3*2, 6*1, etc..

[tex]5x=y+7\Rightarrow \frac{5x}{5}=\frac{y+7}{5}\Rightarrow x=\frac{y+7}{5}\\Plugging\: in:\:\\\frac{y+7}{5}-y>0\Rightarrow \frac{y+7-5y}{5}>0\Rightarrow \frac{-4y+7}{5}>0\Rightarrow \frac{-4y+7}{5}*5>0*5\\-4y+7>0 *(-1)\Rightarrow 4y-7<0\:y>\frac{7}{4}\therefore y<1.75[/tex]

Since y in this hypothetical case is lesser then let's find x, let's plug in y 1 for a value lesser than 1.75:

Then xy≠6 and no and 8/5 (1.75) is a rational number. What makes false the second statement about consecutive integers.

So this is a Contradiction. (x-y) >0 is not true for 5x=x+7.

2) Consider:

x and y are consecutive integers with the same sign is true.

Algebraically speaking, two consecutive integers with the same sign can be  written as:

[tex]y=x+1[/tex]

Plugging in the first equation (5x=y+7):

5x=x+1+7⇒4x=8 ⇒x =2

Since y=3 then x=2 because:

[tex]3=x+1\\3-1=x+1-1\\2=x \Rightarrow x=2[/tex]

3) Testing it

[tex]5x=y+7\\\\5(2)=(3)+7\\\\10=10\:True[/tex]

[tex]xy=6\\2*3=6\\6=6[/tex]

Multiply the cost per yard by the number of yards bought.

Change 3 1/2 to an improper fraction:

3 1/2 = 7/2

Now you have 7/2 x 5/8

Multiply the top numbers together and then the bottom numbers together:

(7 x 5) / (2 x 8) = 35/16

Now rewrite the improper fraction as a proper fraction:

2 3/16 which would be $2.19

Answer:

$12

Step-by-step explanation:

The equation here is (18*2/3).

18 divided by 3 is 6.

6 multiplied by 2 is 12.

Hence, the answer is 12.

Answer:

Interior angle 100 degrees

Exterior angle 80 degrees

Step-by-step explanation:

we know that

The sum of an exterior angle of a triangle and its adjacent interior angle is 180 degrees.

we have that

[tex]10n+(5n+30)=180[/tex]

solve for n

[tex]10n+5n=180-30[/tex]

[tex]15n=150[/tex]

[tex]1n=10[/tex]

Find the measure of the interior angle

[tex]10n=10(10)=100^o[/tex]

Find the measure of the exterior angle

[tex](5n+30)=5(10)+30=80^o[/tex]

Answer:Area of the shaded region is 73.6 cm^2

Step-by-step explanation:

The circle is divided into two sectors. The Smaller sector contains the triangle. The angle that the smaller sector subtends at the center of the circle is 80 degrees. Since the total angle at the center of the circle is 360 degrees, it means that the angle that the larger sector subtends at the center would be 360 - 80 = 280 degrees

Area of a sector is expressed as

Area of sector = #/360 × πr^2

# = 280

r = 5 cm

Area of sector = 280/360 × 3.14 × 5^2

Area of sector = 61.06 cm^2

Area of the triangle is expressed as

1/2bh = 1/2 × 5 × 5 = 12.5

Area of the shaded region = 61.06 +

12.5 = 73.6

Answer:

The correct option is D.

Step-by-step explanation:

It is given that a set of 15 different integers has median of 25 and a range of 25.

Median = 25

Median is the middle term of the data. Number of observations is 15, which is an odd number so median is

[tex](\frac{n+1}{2})th=(\frac{15+1}{2})th=8th[/tex]

8th term is 25. It means 7 terms are less than 25. Assume that those 7 numbers are 18, 19, 20, 21, 22, 23, 24. Largest possible minimum value of the data is 18.

Range = Maximum - Minimum

25 = Maximum - 18

Add 18 on both sides.

25+18 = Maximum

43 = Maximum

The greatest possible integer in this set 43.

Therefore, the correct option is D.

Answer: 15!  or 1307674368000

Step-by-step explanation:

According to the permutations , if we arrange n things in order , then the total number of ways to arrange them = n!

Similarly , when health inspector inspects 15 restaurants in town once each week, the number of different orders can be made for these inspections = 15!

= 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

=1307674368000

Hence, the number of different orders can be made for these inspections = 15! =1307674368000

The number of different orders in which a health inspector can visit 15 restaurants in a week is calculated by computing 15 factorial (15!), resulting in 1,307,674,368,000 different permutations.

The question pertains to the concept of permutations where one is required to determine the number of different orders in which a series of events can occur without repetition. Since the health inspector has to visit 15 different restaurants without visiting the same one more than once in a week, we are dealing with permutations of distinguishable outcomes without repetition where all outcomes are selected. The formula for permutation is n! (n factorial), where n is the number of items to permute. In this case, n is 15 (the number of restaurants).

To calculate the number of different orders for these inspections, you would compute 15!, which is 15 x 14 x 13 x ... x 1. This calculation results in 1,307,674,368,000 different orders in which the health inspector can visit the 15 restaurants. Note that a factorial is the product of all positive integers less than or equal to n. Such permutations ensure that each restaurant is visited once and only once each week, which aligns with professional standards for comprehensive inspections.

Answer:it sent 6945 during the 30 day marketing campaign

Step-by-step explanation:

Their goal is to increase the number of emails sent per day by 15 each day. The rate at which they increased the number of mails sent is in arithmetic progression.

The formula for determining sum of n terms of an arithmetic sequence is expressed as

Sn = n/2[2a + (n - 1)d]

Where

a represents the first term of the sequence.

n represents the number of terms.

d = represents the common difference.

From the information given

a = 14

d = 15

n = 30

We want to find the sum of 30 terms, S30. It becomes

S30 = 30/2[2 × 14 + (30 - 1)15]

S30 = 15[28 + 435]

S30 = 6945