Use logs to determine the number of years until the population drops to 15,390 organisms. Round answer to 2 decimal places.

Answer:

[tex]f(x)=3^{3x-1}[/tex].

The domain of the function is the set of all real number and the range is [tex](0,\infty)[/tex]

Step-by-step explanation:

Given:

The function is  given as:

[tex]f(x)=\frac{1}{3}(81)^{\frac{3x}{4}}[/tex]

Using the rule of the exponents, [tex]a^{mn}=(a^m)^n[/tex],

[tex]f(x)=\frac{1}{3}((81)^{\frac{1}{4}})^{(3x)}\\f(x)=\frac{1}{3}(\sqrt[4]{81} )^{3x}\\f(x)=\frac{1}{3}(3)^{3x}\\f(x)=\frac{3^{3x}}{3^1}[/tex]

Using the rule of the exponents,[tex]\frac{a^m}{a^n}=a^{m-n}[/tex],

[tex]f(x)=3^{3x-1}[/tex]

Therefore, the simplified form of the given function is:

[tex]f(x)=3^{3x-1}[/tex]

Key aspects:

The given function is an exponential function with a constant base 3.

Domain is the set of all possible values of [tex]x[/tex] for which the function is defined.

The domain of an exponential function is a set of all real values.

The range of an exponential function is always greater than zero.

Therefore, the domain of this function is also all real values and the range is from 0 to infinity.

Domain: [tex]x \epsilon (-\infty,\infty)[/tex]

Range: [tex]y\epsilon (0,\infty)[/tex]

Answer: 1/3 27 all real numbers y>0

Step-by-step explanation:

Her total sales was $3695.33

Step-by-step explanation:

Osmin had a gross pay of $624.86 last week

She wants to know what was her total sales

Assume that her total sales is $x

∵ Her gross pay = $624.86

∵ She earns 12.85 per hour plus a 3% commission on all sales

∵ She worked for 40 hrs last week

∵ Her total sales was $x

- Put all of these in an equation represents her gross pay

∵ Her gross pay = her rate per hour × the number of hours + 3% of x

∴ 624.86 = 12.85 (40) + 3% (x)

∴ 624.86 = 514 + [tex]\frac{3}{100}[/tex] x

∴ 624.86 = 514 + 0.03 x

- Subtract 514 from both sides

∴ 110.86 = 0.03 x

- Divide both sides by 0.03

∴ 3695.33 = x

∴ Her total sales = $3695.33

Her total sales was $3695.33

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Answer:

Spearman's correlation

Step-by-step explanation:

A researcher records each student’s rank in his/her high school graduating class and the student’s rank in the college graduating class.

The correlation that should be used to measure the relationship between these two variables is - Spearman's correlation

This correlation gives a statistical measure of similar relationship between paired data.

This is used to evaluate relationships involving ordinal variables.

Answer:

The Total distance she travel fro work is 30 miles  .

Step-by-step explanation:

Given as :

the total time taken to cover distance = 45 minutes

Let The total distance cover = D miles

The distance cover at the speed of 32 mph = [tex]D_1[/tex] miles

The time taken to cover [tex]D_1[/tex] miles distance = [tex]\frac{1}{2}[/tex] hour

Distance = Speed × Time

∴  [tex]D_1[/tex] = 32 mph  ×  [tex]\frac{1}{2}[/tex] h

or, [tex]D_1[/tex] = 16 miles

Again ,

The distance cover at the speed of 56 mph = [tex]D_2[/tex] miles

The time taken to cover [tex]D_2[/tex] miles distance =  [tex]\frac{1}{4}[/tex] hour

∴  [tex]D_2[/tex] = 56 mph  ×  [tex]\frac{1}{4}[/tex] h

or, [tex]D_2[/tex] = 14 miles

So , The total distance she travel for work =  [tex]D_1[/tex]  +  [tex]D_2[/tex]

Or, The total distance she travel for work = 16 miles + 14 miles = 30 miles

Hence The Total distance she travel fro work is 30 miles  . Answer

Answer:

Step-by-step explanation:

For given points (t1, y1), (t2, y2), I like to write the exponential function as ...

  y(t) = y1·(y2/y1)^((t-t1)/(t2-t1))

This can be converted to other forms (such as a·b^t or a·e^(kt)) fairly easily, but those tend not to reproduce the given numbers exactly as this form does.

Using (15, 300) and (35, 1900) as our data values, the exponential function can be written as ...

  y(t) = 300·(19/3)^((t-15)/20)

__

a) The initial size of the culture is the value of y(0).

 y(0) = 300·(19/3)^(-15/20) ≈ 75.144

  y(0) ≈ 75 . . . initial population

__

b) The doubling period will be the value of t that satisfies ...

  (19/3)^(t/20) = 2

Taking logarithms, we have ...

  (t/20)·log(19/3) = log(2)

  t = 20·log(2)/log(19/3) ≈ 7.5104 . . . . minutes

The doubling time is about 7.51 minutes.

__

c) Evaluating the formula for t=115, we have ...

  y(115) = 300·(19/3)^(100/20) ≈ 3056912.346

The count after 115 minutes will be about 3,056,900.

__

d) Solving y(t) = 11,000, we have ...

  11000 = 300·(19/3)^((t-15)/20)

  11000/300 = (19/3)^((t-15)/20)

  log(110/3) = (t-15)/20·log(19/3)

  t = 20·log(110/3)/log(19/3) + 15 ≈ 54.027

It will take about 54.03 minutes for the count to reach 11,000.

_____

I find a graphing calculator to be a nice tool for solving problems like this.

Answer: The clock skip 0.25 seconds every 30 minutes.

Step-by-step explanation:

Since we have given that

Clock was correct on Wednesday at 4 pm.

At 2 pm , on saturday, the clock was running late by 35 seconds.

From 4 pm wednesday to 4 pm thursday = 24 hours

From 4 pm thursday to 4 pm friday = 24 hours

From 4 pm friday to 2 pm saturday = 22 hours

So, total hours = 24+24+22 = 70 hours

We need to find the number of seconds that the clock skip every 30 minutes.

So, it becomes

[tex]\dfrac{70}{0.5}=\dfrac{35}{T}\\\\70T=35\times 0.5\\\\T=\dfrac{35\times 0.5}{70}\\\\T=0.25[/tex]

Hence, the clock skip 0.25 seconds every 30 minutes.

The clock was 35 seconds late over a period of 70 hours, which equals 140 intervals of 30 minutes. Therefore, the clock skipped an average of 0.25 seconds every 30 minutes.

To determine the average number of seconds the clock skipped every 30 minutes, we first need to calculate the total time difference and then divide by the number of 30-minute intervals.

The clock was showing the correct time on Wednesday at 4 pm.

It was 35 seconds late by Saturday at 2 pm.

Time elapsed from Wednesday 4 pm to Saturday 2 pm is 2 days and 22 hours, which equals (2×24 + 22) hours = 70 hours.

Converting 70 hours into minutes: 70×60 = 4200 minutes.

Number of 30-minute intervals in this period: 4200 / 30 = 140 intervals.

Average seconds skipped per 30-minute interval: 35 seconds / 140 intervals = 0.25 seconds.

Thus, the clock skipped an average of 0.25 seconds every 30 minutes.

Answer:

0.9953, 3.3629<x<4.0371

Step-by-step explanation:

Given that slader the internal revenue service claims it takes an average of 3.7 hours to complete a 1040 tax form, assuming th4e time to complete the form is normally distributed witha standard devait of the 30 minutes:

If X represents the time to complete then

X is N(3.7, 0.5) (we convert into uniform units in hours)

a) percent of people would you expect to complete the form in less than 5 hours

=[tex]100*P(x<5)\\= 0.9953[/tex]

b) P(b<x<c) = 0.50

we find that here

c = 4.0371 and

b = 3.3629

Interval would be

[tex](3.3629, 4.0371)[/tex]

Answer:

Yes, it will reach or exceed 141 degree F

Step-by-step explanation:

Given equation that shows the temperature T in degrees Fahrenheit x minutes after the machine is put into operation is,

[tex]T = 0.005x^2 + 0.45x + 125[/tex]

Suppose T = 141°F,

[tex]\implies 141 = 0.005x^2 + 0.45x + 125[/tex]

[tex]\implies 0.005x^2 + 0.45x + 125 - 141 =0[/tex]

[tex]\implies 0.005x^2 + 0.45x - 16=0[/tex]

Since, a quadratic equation [tex]ax^2 + bx + c =0[/tex] has,

Real roots,

If Discriminant, [tex]D = b^2 - 4ac \geq 0[/tex]

Imaginary roots,

If D < 0,

Since, [tex]0.45^2 - 4\times 0.005\times -16 = 0.2025 + 32 > 0[/tex]

Thus, roots of -0.005x² + 0.45x + 125 are real.

Hence, the temperature can reach or exceed 141 degree F.

The temperature of the part will exceed 141°F during the manufacturing process.

To determine if the temperature of the part will ever reach or exceed 141°F, we need to find the value of x when the temperature T equals 141°F. We can do this by setting the equation T = 0.005x² + 0.45x + 125 equal to 141 and solving for x using the quadratic formula.

The quadratic formula is given by x = (-b ± √(b² - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = 0.005, b = 0.45, and c = 125 - 141 = -16.

Calculating the discriminant, which is the value inside the square root in the quadratic formula, we get b² - 4ac = 0.45² - 4(0.005)(-16) = 0.2025 + 0.32 = 0.5225. Since the discriminant is positive, the quadratic equation has two real and distinct solutions, which means the temperature of the part will exceed 141°F at some point during the manufacturing process.

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Answer:

segment

Step-by-step explanation:

We know that a line has no end points.

If we take a part from the line then it is called line segment.

The line segment has starting point and end point.

A part of a line consisting of two endpoints and all the points between them is called segment.

Therefore, the answer is segment

Line segment

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Answer:

The answers should be c. 120° and e. 240°

Step-by-step explanation:

Consider the provided information.

Equilateral triangle has 3 equal sides.

Now we need to rotate the equilateral triangle so that the equilateral triangle to map it onto itself.

For this we need to rotate each of those sides to an adjacent side. (Shown in figure)

This can be happen 3 times as there are 3 sides,

A circle has 360° and [tex]\dfrac{1}{3}\times 360^{\circ}=120^{\circ}[/tex].

Thus, a 120° rotation map it onto itself.

Any other angle multiple of 120° will do the same.

Hence, the answers should be c. 120° and e. 240°

Answer:

D. Reject the null hypothesis as the test statistic is greater than the critical value of z.

Step-by-step explanation:

[tex]H_{0}:[/tex] welders earn $50,000 annually

[tex]H_{a}:[/tex] welders' income does not equal $50,000 annually

Sample size 100>30, therefore we need to calculate z-values of sample mean and significance.

z-critical at 0.10 significance is 1.65

z-score of sample mean (test statistic) can be calculated as follows:

[tex]\frac{X-M}{\frac{s}{\sqrt{N} } }[/tex] where

Then z=[tex]\frac{50,350-50,000}{\frac{2,000}{\sqrt{100} } }[/tex] =

1.75.

Since test statistic is bigger than z-critical, (1.75>1.65), we reject the null hypothesis.

To answer the question, a z-test was performed comparing the sample mean income of welders to the population mean. The test statistic (1.75) was found to be greater than the critical z-value (±1.645) for a 0.10 alpha level, therefore we reject the null hypothesis.

The question is asking us to conduct a hypothesis test to determine whether shipbuilders' welders earn more or less than the population mean income of certified welders, which is $50,000. In statistics, we normally use a z-test for such a comparison when we know the population standard deviation. Given that the level of significance (alpha) is 0.10, we must find the critical z-value that corresponds to this alpha level, calculate the test statistic for the sample mean of $50,350, then compare our test statistic with the critical value to make our decision.

Since we have a large sample size (n=100) and the population standard deviation is known, a z-test is appropriate. The test statistic is calculated using the formula:

z = (X_bar- μ) / (σ/√n)

Where X_bar is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging in the values:

z = ($50,350 - $50,000) / ($2,000/√100) = $350 / $200 = 1.75

To determine whether to reject the null hypothesis, we must compare the test statistic to the critical value of z for a significance level of 0.10. For a two-tailed test (since we want to know if it's more or less, not just more), the critical z-values are approximately ±1.645. Since our test statistic of 1.75 is greater than 1.645, we reject the null hypothesis, implying that there is enough evidence to suggest the mean annual income of the sample of welders is different from $50,000. However, as the question does not specify the direction of the alternative hypothesis (whether we were testing for higher or lower earnings, specifically), we cannot conclude that welders earn more than $50,000 without further information.

Both jewelers used 26104 beads altogether while making necklaces.

Step-by-step explanation:

No. of necklaces made by Gemma = 106

Necklaces made by Leah = 106+39 = 145 necklaces

Total necklaces made = Gemma's + Leah's

Total necklaces made = [tex]106+145 = 251\ necklaces[/tex]

Beads used in 1 necklace = 104 beads

Beads used in 251 necklaces = 104*251 = 26104

Both jewelers used 26104 beads altogether while making necklaces.

Keywords: multiplication, addition

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Answer:

C

Step-by-step explanation:

(2x + 3)^5 = C(5,0)2x^5*3^0 +

C(5,1)2x^4*3^1 + C(5,2)2x^3*3^2 + C(5,3)2x^2*3^3 + C(5,4)2x^1*3^4 + C(5,5)2x^0*3^5

Recall that

C(n,r) = n! / (n-r)! r!

C(5,0) = 1

C(5,1) = 5

C(5,2) = 10

C(5,3) = 10

C(5,4) = 5

C(5,5) = 1

= 1(2x^5)1 + 5(2x^4)3 + 10(2x^3)3^2 + 10(2x^2)3^3 + 5(2x^1)3^4 + 1(2x^0)3^5

= 2x^5 + 15(2x^4) + 90(2x^3) + 270(2x^2) + 405(2x) +243

= 32x^5 + 15(16x^4) + 90(8x^3) + 270(4x^2) + 810x + 243

= 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243

Answer:

the answer is C

Step-by-step explanation:

Answer:

Volume of wax for 210 candles=18463.2 cubic inches

Step-by-step explanation:

Each wax candle has a cylindrical shape with

Number of candles to be made=210

Volume of one wax candle =π × [tex]r^{2}[/tex] × h

                                             =π ×[tex]2^{2}[/tex] × 7

                                             =3.14×4×7

                                             =87.92 cubic inches

Volume of 210 wax candles=210×87.92

                                              =18,463.2 cubic inches

Answer:

18,471.6

Step-by-step explanation:

Answer:

The answer to your question is 43

Step-by-step explanation:

                  2x³  + 8x² - 5x + 5 / x - 2

Process

1.- Use synthetic division

                      2   8    -5    5     2

                           4    24  38                          

                      2  12    19   43

Quotient    2x² + 12x + 19

Remainder     43

Answer: 70

Step-by-step explanation:

8x+6=4x+38

4x=32

x=8

<B=4x+38=4*8+38=70

Answer:

x = 8

∠B = 70°

Step-by-step explanation:

∠A = ∠B through alternate exterior angles.

∠A = ∠B

8x + 6 = 4x + 38

8x - 4x + 6 = 38

4x + 6 = 38

4x = 38 - 6

4x = 32

x = 32 ÷ 4

x = 8

∠B = 4x + 38

4(8) + 38

32 + 38

= 70

Answer:

28.27 cm/s

Step-by-step explanation:

Though Process:

Solution:

the area of the exposed circle is

[tex]A =\pi a^2 [/tex]

the rate of change of this area can be, (using chain rule)

[tex]\frac{dA}{dt} = 2 \pi a \frac{da}{dt}[/tex] we can call this Eq(2)

what we are really concerned about is how [tex]a[/tex] changes as the punch is being poured into the bowl i.e [tex]\frac{da}{dh}[/tex]

So we need another formula: Using the property of hemispheres and pythagoras theorem, we can use:

[tex]r = \frac{a^2 + h^2}{2h}[/tex]

and rearrage the formula so that a is the subject:

[tex]a^2 = 2rh - h^2[/tex]

now we can derivate a with respect to h to get [tex]\frac{da}{dh}[/tex]

[tex]2a \frac{da}{dh} = 2r - 2h[/tex]

simplify

[tex]\frac{da}{dh} = \frac{r-h}{a}[/tex]

we can put this in Eq(1) in place of [tex]\frac{da}{dh}[/tex]

[tex]\frac{da}{dt} = \frac{r-h}{a} . \frac{dh}{dt}[/tex]

and since we know [tex]\frac{dh}{dt} = 1.5[/tex]

[tex]\frac{da}{dt} = \frac{(r-h)(1.5)}{a} [/tex]

and now we use substitute this [tex]\frac{da}{dt}[/tex]. in Eq(2)

[tex]\frac{dA}{dt} = 2 \pi a \frac{(r-h)(1.5)}{a}[/tex]

simplify,

[tex]\frac{dA}{dt} = 3 \pi (r-h)[/tex]

This is the rate of change of area, this is being asked in the quesiton!

Finally, we can put our known values:

[tex]r = 5cm[/tex]

[tex]h = 2cm[/tex] from the question

[tex]\frac{dA}{dt} = 3 \pi (5-2)[/tex]

[tex]\frac{dA}{dt} = 9 \pi cm/s// or//\frac{dA}{dt} = 28.27 cm/s[/tex]

Answer:

The over payment amount after 36 months is $ 0.37

Step-by-step explanation:

Given as :

The cost of the Car = $ 26,555

The number of times payment done for 36 month = 36

So, The the cost of car for 36 equal payment = [tex]\dfrac{\textrm Total cost of car}{\textrm number of times payment done }[/tex]

i.e The the cost of car for 36 equal payment = [tex]\frac{26,555}{36}[/tex]

∴ The the cost of car for 36 equal payment = $ 737.6389

Now rounding this value to nearest whole dollar = $ 738

Note - A) If after decimal , number are above 4 then round it to 1 above digit

B) If after decimal , number 4 or less then simply remove all number   after       decimal

So, The over payment after 36 months = $ 738 - $ 737.63 = $ 0.37

Hence The over payment amount after 36 months is $ 0.37   Answer

Answer:

The mean of a binomial distribution is given  by  mean  = n x p where n = the number of items and p equals the probability of success.  Here we have:

mean =  1632 x 0.57  =   930.2

Step-by-step explanation:

The mean for a binomial substitution = n x p

Mean = 1632 x 0.57 = 930.24

The answer would be D.

Picture:

Multiply P(x) by X, then add those together:

0 x 0.42 = 0

1 x 0.12 = 0.12

2 x 0.34 = 0.68

3 x 0.05 = 0.15

4 x 0.07 = 0.28

Mean = 0 + 0.12 + 0.68 + 0.15 + 0.28 = 1.23

The function f(x)=ln(9.2x) is a horizontal compression of the parent function by a factor of 1/9.2

Step-by-step explanation:

The multiplication of a function by a number compresses or stretches the function vertically while to compress or stretch the function horizontally, the input variable is multiplied with a number.

i.e.

[tex]For\ f(x) => g = f(bx)[/tex]

where b is a constant.

Now

If b>0 then the function is compressed horizontally

The given function is:

[tex]f(x) = ln\ x\\Transformed\ to\\f(x) = ln\ (9.2x)[/tex]

As the variable in function is multiplied with a number greater than zero, the function will stretch horizontally.

The function f(x)=ln(9.2x) is a horizontal compression of the parent function by a factor of 1/9.2

Keywords: Transformation

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Answer:

The function f(x)=In (9.2x) is a horizontal compression of the parent function by a factor of 5/46

Step-by-step explanation:

I just took this test and that was the correct answer :( good luck everyone

Answer:

  12 mph

Step-by-step explanation:

The relationship between jogging speed and walking speed means the time it takes to walk 4 miles is the same as the time it takes to jog 8 miles. Then the total travel time (0.75 h) is the time it would take to jog 1+8 = 9 miles. The jogging speed is ...

  (9 mi)(.75 h) = 12 mi/h . . . average jogging speed

__

Check

1 mile will take (1 mi)/(12 mi/h) = 1/12 h to jog.

4 miles will take (4 mi)/(6 mi/h) = 4/6 = 8/12 h to walk.

The total travel time is (1/12 +8/12) h = 9/12 h = 3/4 h. (answer checks OK)

_____

Comment on the problem

Olympic race-walking speed is on the order of 7.7 mi/h, so John's walking speed of 6 mi/h should be considered quite a bit faster than normal. The fastest marathon ever run is on the order of a bit more than 12 mi/h, so John's jogging speed is also quite a bit faster than normal. No wonder he got tired.

Answer:

16.78 seconds

Step-by-step explanation:

speed = Distance Traveled / time

thus speed =29.79 Km/sec

time =distance Traveled/speed (from above formula)

time taken=500 km ÷ 29.79 Km/sec

∴time taken=16.78 seconds

Answer: length of the drive way = 56 feet

Width of the driveway = 12 feet

Step-by-step explanation:

The rectangular driveway has two equal lengths and two equal widths. The area of the driveway is expressed as

length,l × width,w

The area is 672 feet squared. It means that

L×W = 672

The length of the rectangular driveway is four feet less than five times the width. It means that

L = 5W - 4

Substituting L = 5W - 4 into LW = 672

W(5W - 4) = 672

5W^2 - 4W - 672 = 0

5W^2 + 56W - 60W - 672 = 0

W(5W + 56) - 12(5W + 56) = 0

(W - 12)(5W + 56) = 0

W - 12 = 0 or 5W + 56 = 0

W = 12 or 5W = -56

W= 12 or W = - 56/5

The Width cannot be negative , so

W = 12

LW = 672

12L = 672

L = 672/12 = 56

Time taken by jerry alone is 10.1 hours

Time taken by callie alone is 8.1 hours

Solution:

Given:- For a certain prospectus Callie can prepare it two hours faster than Terry can

Let the time taken by Terry be "a" hours

So, the time taken by Callie will be (a-2) hours

Hence, the efficiency of Callie and Terry per hour is [tex]\frac{1}{a-2} \text { and } \frac{1}{a} \text { respectively }[/tex]

If they work together they can do the entire prospectus in five hours

[tex]\text {So, } \frac{1}{a-2}+\frac{1}{a}=\frac{1}{5}[/tex]

On cross-multiplication we get,

[tex]\frac{a+(a-2)}{(a-2) \times a}=\frac{1}{5}[/tex]

[tex]\frac{2 a-2}{(a-2) \times a}=\frac{1}{5}[/tex]

On cross multiplication ,we get

[tex]\begin{array}{l}{5 \times(2 a-2)=a \times(a-2)} \\\\ {10 a-10=a^{2}-2 a} \\\\ {a^{2}-2 a-10 a+10=0} \\\\ {a^{2}-12 a+10=0}\end{array}[/tex]

using quadratic formula:-

[tex]x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]

[tex]x=\frac{12 \pm \sqrt{144-40}}{2}[/tex]

[tex]\begin{array}{l}{x=\frac{12 \pm \sqrt{144-40}}{2}} \\\\ {x=\frac{12 \pm \sqrt{104}}{2}} \\\\ {x=\frac{12 \pm 2 \sqrt{26}}{2}} \\\\ {x=6 \pm \sqrt{26}=6 \pm 5.1} \\\\ {x=10.1 \text { or } x=0.9}\end{array}[/tex]

If we take a = 0.9, then while calculating time taken by callie = a - 2 we will end up in negative value

Let us take a = 10.1

So time taken by jerry alone = a = 10.1 hours

Time taken by callie alone = a - 2 = 10.1 - 2 = 8.1 hours

Answer:

Step-by-step explanation:

We want to find 95% confidence interval for the mean of the weight of of textbooks.

Number of samples. n = 28 textbooks weight

Mean, u =76 ounces

Standard deviation, s = 12.3 ounces

For a confidence level of 95%, the corresponding z value is 1.96. This is determined from the normal distribution table.

We will apply the formula

Confidence interval

= mean +/- z ×standard deviation/√n

It becomes

76 +/- 1.96 × 12.3/√28

= 76 +/- 1.96 × 2.3113

= 76 +/- 4.53

The lower end of the confidence interval is 76 - 4.53 =71.47

The upper end of the confidence interval is 76 + 4.53 = 80.53

Therefore, with 95% confidence interval, the mean textbook weight is between 71.47 ounces and 80.53 ounces

Answer:

Maximum = 540 at (6,14)

Minimum = 300 at (0,10) or (12,2).

Step-by-step explanation:

The given linear programming problem is

Minimize and maximize: P = 20x + 30y

Subject to constraint,

[tex]2x+3y\ge 30[/tex]            .... (1)

[tex]2x+y\le 26[/tex]            .... (2)

[tex]-2x+3y\le 30[/tex]            .... (3)

[tex]x,y\geq 0[/tex]

The related equation of given inequalities are

[tex]2x+3y=30[/tex]

[tex]2x+y=26[/tex]

[tex]-2x+3y=30[/tex]

Table of values are:

For inequality (1).

x      y

0     10

15     0

For inequality (2).

x      y

0     26

13     0

For inequality (3).

x      y

0     10

15     0

Pot these ordered pairs on a coordinate plane and connect them draw the corresponding related line.

Check each inequality by (0,0).

[tex]2(0)+3(0)\ge 30\Rightarrow 0\ge 30[/tex]    False

[tex]2(0)+(0)\le 26\Rightarrow 0\le 26[/tex]     True

[tex]-2(0)+3(0)\le 30\Rightarrow 0\le 30[/tex]    True

It means (0,0) is included in the shaded region of inequality (2) and (3), and (0,0) is not included in the shaded region of inequality (1).

From the below graph it is clear that the vertices of feasible region are (0,10), (6,14) and (12,2).

Calculate the values of objective function on vertices of feasible region.

Point           P = 20x + 30y

(0,10)           P = 20(0) + 30(10) = 300

(6,14)           P = 20(6) + 30(14) = 540

(12,2)           P = 20(12) + 30(2) = 300

It means objective function is maximum at (6,14) and minimum at (0,10) or (12,2).

Let the number of bikes = x and total money = y.

Set  up two equations:

The first equation is the cost function, which would be rent plus 60 times the number of bikes:

y = 1800 + 60x

The second equation would be revenue, where total money would be equal to 120 times the number of bikes sold:

y = 120x

Now you can graph both equations by setting the equations to equal each other. The point on the graph, where the loib=ne crosses the X axis is the break even point

Graph

120x = 1800 +60x

See attached picture for the graph and you can see the break even point is 30 bikes.

You can check by replacing X with 30 to see if the equations equal each other:

1800 + 60(30) = 1800 +1800 = 3600

120(30) = 3600

Answer:

The number of pitchers produced by each container = 16 .

Step-by-step explanation:

Given,

⇒∴ The number of cups of mix in brand A = [tex]x+4[/tex];

Number of pitchers produced by the containers :

Since both are equal:

⇒[tex]2x+8 = 4x\\8=2x\\x=4[/tex]

Thus the number of cups of mix in Brand B = [tex]x=4[/tex];

The number of pitchers produced by each container :

= [tex]\frac{4}{\frac{1}{4} } \\= 4*4\\=16[/tex]

∴The number of pitchers produced by each container = 16.

Each container can make 16 pitchers.

In mathematics, a fraction is used to denote a portion or component of the whole. It stands for the proportionate pieces of the whole.

As per the given data:

For making a pitcher of brand, A 1/2 cup of a mix is required.

For making a pitcher of brand, B 1/4 cup of a mix is required.

For brand A, 4 more cups than brand B .

Let's assume the number of cups for brand B as x

∴ Number of cups for brand A = x + 4

Total number of pitchers = Total number of cups / cups for one pitcher

For brand A number of pitchers:

= [tex]\frac{x + 4}{\frac12}[/tex] = 2(x + 4)

For brand B number of pitchers:

= [tex]\frac{x}{\frac14}[/tex] = 4x

The number of pitchers will be same for both brand A and B

∴ 2(x + 4) = 4x

= 2x + 8 = 4x

x = 4

The number of pitchers = [tex]\frac{4}{\frac14}[/tex] = 16

Hence, each container can make 16 pitchers.

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Amount of down payment = $46000

Mortgage needs = $184000

Solution:

From the given,

Cost of the house = [tex]\$230000[/tex]

Percentage of down payment = [tex]20\%[/tex]

Number of years of fixed loan = 30

[tex]\text { Total down payment }=\text { cost of the house } \times \text { Percentage of down payment }[/tex]

[tex]\Rightarrow \frac{230000 \$\times 20}{100} \rightarrow 46000 \$[/tex]

[tex]\text {Mortgage needs}=\text { Total cost - Total down payment }[/tex]

[tex]\Rightarrow 230000 \$-46000 \$=184000 \$[/tex]

It can be concluded that the total down payment for the house and mortgage needs would be [tex]\$46000 \text{ and } \$184000[/tex]

Given a ratio of cheese to turkey slices is 2:3, Mr. Lynch spent $18 on cheese and $27 on turkey slices out of a total of $45.

Mr. Lynch is faced with a real-world example of the concept of ratios. In this case, the ratio of cheese to turkey slices is 2:3, meaning for every 2 parts of cheese, he's spending on 3 parts of turkey slices. The total amount spent is $45. We can solve this ratio problem to identify the individual costs for cheese and turkey slices by using a simple mathematical method.

Firstly, let's understand how a ratio works. It's a way to compare amounts of different things. Given the ratio 2:3 (cheese to turkey), add the ratio numbers together to get the total parts, i.e., 2 + 3 = 5. These 5 parts represent the total amount of $45 spent.

Next, we need to find the value of one part. To do this, divide the total amount spent by the total parts: $45/5 parts gives us 9: this determines that each part is worth $9.

Finally, to find the amounts spent on cheese and turkey slices, multiply the number of parts each item has in the ratio by the value of one part. So, for cheese, it's 2 parts x $9 = $18, and for turkey slices, it's 3 parts x $9 = $27.

In conclusion, Mr. Lynch spent $18 on cheese and $27 on turkey slices.

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