What is the solution to the following system of equations?

x − 4y = 6
2x + 2y = 12

answer choices
(0,10)
(10,0)
(6,0)
(0,6)

Answer: 126.76056338%

Step-by-step explanation: To answer the question what percent of 71 is 90, we translate the question into an equation.

We have "what percent" which we can represent by the variable X. Next we have "of 71" which means "times 71" and "is 90" means "equals 90."

Now to solve for x, since x is being multiplied by 71, we need to divide by 71 on both sides of the equation.

On the left side of the equation the 71's cancel and we have x. On the right side of the equation we have 90 divided by 71 which is 1.2676056338.

Now, remember that we want to write our answer as a percent. To write 1.2676056338 as a percent, we move the decimal point two places to the right and we get 126.76056338%.

Answer:

78.89%

Step-by-step explanation:

Final answer:

The amount of rain that fell on the banana farm in April is 153 millimeters.

Explanation:

To find the amount of rain that fell on the banana farm in April, subtract the amount of rain that fell in March from the total rainfall. The total rainfall in March and April is 271 millimeters, and the amount of rain that fell in March is 118 millimeters. So, the amount of rain that fell in April is 271 millimeters - 118 millimeters = 153 millimeters.

The right answer is Option C.

Step-by-step explanation:

Given,

2/5 cup of coconut shavings covers 1/4 of a cake,

[tex]\frac{2}{5}\ coconut\ shavings=\frac{1}{4}\ of\ cake[/tex]

For one whole cake, we will multiply both sides with 4;

[tex]4*\frac{2}{5}=\frac{1}{4}*4\\\frac{8}{5} = 1\ cake[/tex]

Writing in simplified fraction form;

[tex]1\ cake = 1\frac{3}{5}\ cups[/tex]

The right answer is Option C.

Keywords: fractions, multiplication

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Answer:r = 2.81 centimeters

Step-by-step explanation:

The area of this circle can be found using the formula

A=πr^2

Where r = radius of the circle

π is a constant whose value is 3.14

We are given the area of the circle to be 25 square centimeters. To determine the length of the radius of the circle , we will make r the subject of the formula

r^2 = A/π

Taking square root of both sides of the equation,

r = √A/π

Since A = 25 and π = 3.14,

r= √25/3.14

r = 2.82166323992

Approximating to the nearest tenth of a centimeter,

r = 2.81 centimeters

Answer:

Part a) [tex]120\pi\ \frac{rad}{min}[/tex]

Part b) [tex]960\pi\ \frac{in}{min}[/tex]

Step-by-step explanation:

we have

60 rev/min

Part a) Find the angular speed in radians per minute

we know that

One revolution represent 2π radians (complete circle)

so

[tex]1\ rev=2\pi \ rad[/tex]

To convert rev to rad, multiply by 2π

[tex]60\ \frac{rev}{min}=60(2\pi)=120\pi\ \frac{rad}{min}[/tex]

Part b) Find the linear speed in inches per minute

we know that

The circumference of a circle is equal to

[tex]C=2\pi r[/tex]

we have

[tex]r=8\ in[/tex] ----> given problem

substitute

[tex]C=2\pi(8)[/tex]

[tex]C=16\pi\ in[/tex]

Remember that

One revolution subtends a length equal to the circumference of the circle

so

[tex]1\ rev=16\pi\ in[/tex]

To convert rev to in, multiply by 16π

[tex]60\ \frac{rev}{min}=60(16\pi)=960\pi\ \frac{in}{min}[/tex]

Final answer:

The angular speed of the pulley is 120π radians per minute, and the linear speed is 960π inches per minute.

Explanation:

The question pertains to angular and linear speeds related to circular motion. Given that a belt runs a pulley with a radius of 8 inches at 60 revolutions per minute (rpm), we are tasked with finding both the angular speed in radians per minute and the linear speed in inches per minute.

Calculating the Angular Speed

The angular speed (ω) in radians per minute can be calculated using the formula ω = 2π×rpm, where rpm is the number of revolutions per minute and 2π radians is the equivalent of one full revolution.

ω = 2π × 60 = 120π radians/minute

Calculating the Linear Speed

The linear speed (v) can be determined from the radius (r) and angular speed (ω) using the formula v = r×ω. The radius of the pulley is 8 inches, so:

v = 8 inches × 120π radians/minute = 960π inches/minute

Answer:

Step-by-step explanation:

If the width of the original piece of sheet metal is x, then the length is 2x-10. Subtracting a 5" square from each corner makes the bottom of the box have dimensions (x-10) and (2x-20). The volume of the box is then ...

  5(x -10)(2x -20) = 1210 . . . . . . volume is the product of the dimensions

  10(x -10)² = 1210 = 10(11²) . . . . factor the equation

  (x -10)² = 11² . . . . . . . . divide by 10

  x = 11 + 10 = 21 . . . . . .take the square root, add 10

The length and width of the original piece of sheet metal were 32 inches and 21 inches, respectively.

The appropriate null and alternative hypotheses for the situation, the consequences of Type I and Type II errors in this context.

a. Null Hypothesis (H0): The compensation plan does not increase the average sales per salesperson. Alternative Hypothesis (H1): The compensation plan does increase the average sales per salesperson.

b. Type I error occurs when the null hypothesis is rejected, but it is actually true. In this situation, it means concluding that the compensation plan increases the average sales per salesperson when it does not. The consequence is that the company may implement the compensation plan and incur unnecessary costs.

c. Type II error occurs when the null hypothesis is accepted, but it is actually false. In this situation, it means concluding that the compensation plan does not increase the average sales per salesperson when it actually does. The consequence is that the company may miss out on the benefits of the compensation plan and fail to motivate its salespeople effectively.

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Answer:

1.5 hours

Step-by-step explanation:

Since Joe painted 2/3 section (2 out of 3 sections) of each section, we can break up the sections into parts of "3".

So 8 sections would have total subsections of 8 * 3 = 24 subsections

He painted 2 subsections of each of the 8 sections, so number of subsections painted would be:

8 * 2 = 16 subsections

There are more 24 - 16 = 8 more subsections to paint

If 1 hour means 16 subsections,

1/16 hour = 1 subjection

Thus,

8 remaining subsections would take:

8 * 1/16 = 8/16 = 1/2 hour

So

Initially 1 hour, then 1/2 hour, so total of 1 + 1/2 = 1.5 hours

So, to paint all sections, he took 1.5 hours

Answer:

53/72

Step-by-step explanation:

5/8 = x - 1/9

+1/9       +1/9

5/8 + 1/9 = x

9(5/8)  + 8(1/9)

45/72 + 8/72

x= 53/72

The rate of change of the volume of a ball when the radius is 5cm is -300π cubic centimeters per second.

To find the rate of change of the volume of a ball, we can use the formula for the volume of a sphere, which is V = (4/3)πR³. We are given that the radius is decreasing at a constant rate of 3 cm per second. So, the rate of change of the volume can be found using the derivative of the volume function with respect to time.

First, we differentiate the volume function with respect to time:

dV/dt = (4/3)π×3R²×(-3)

dV/dt = -12πR²

Then we substitute the given value of the radius when it is 5 cm:

dV/dt = -12π×5²

dV/dt = -12π×25

dV/dt = -300π

Therefore, the rate of change of the volume of the ball when the radius is 5 cm is -300π cubic centimeters per second.

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Answer:

B, E, F

Step-by-step explanation:

Simply write the terms out and match them.

Final answer:

The correct ordered pairs formed from combining a term in Pattern A with its corresponding term in Pattern B are:

(1, 10), (5, 6), and (7, 4), which are Options B, E, and F, respectively.

Explanation:

The question involves creating ordered pairs from two different number patterns, Pattern A and Pattern B. To find the correct ordered pairs, we match each term in Pattern A with its corresponding term in Pattern B.

To identify the corresponding terms, we look at the positions: the first term in Pattern A pairs with the first in Pattern B, the second with the second, and so on. Here are the resulting pairs:

Based on the above matching, we evaluate the given options:

Therefore, the correct ordered pairs when combining a term from Pattern A with its corresponding term from Pattern B are Option B, Option E, and Option F.

Answer:

 h  =   1,743

Step-by-step explanation:

Volume of a box is

V(h) =  ( A  -  2h) *  ( B  -  2h)* h              A  =  7     B  =  12

We have

V(h) = ( 7  -  2h)  *  ( 12  -  2h ) * h

V(h)  = ( 84  -  14*h  -  24*h  + 4*h² )  * h

V(h)  =  ( 84 -  38*h  + 4 *h² ) * h     ⇒  V(h)  =  84h - 38h² + 4h³

Taking derivatives both sides of the equation

V´(h)    =  84 - 76h + 12x²

V´(h)    =  0              84 - 76h + 12x² = 0     42 - 38h + 6x²

3x² - 19h   + 24  = 0

Solving for h          h1  = [  ( 19 + √(19)² - 288  ]/ 6    h1  = [ (19 + √73)/6]

h₁  =  4,59   we dismiss this value since 9,18  (4,59*2)  > A

h₂  = [ 19 - √73)/6]          h₂  =   1,743

h  =  1.743 is h value to maximizes V

Answer:

350

Step-by-step explanation:

If x is the number of hamburgers and y is the number of cheeseburgers, then:

x + y = 764

y = x + 64

Solve the system of equations with substitution or elimination.  Using substitution:

x + x + 64 = 764

2x + 64 = 764

2x = 700

x = 350

350 hamburgers were sold.

Answer:

The probability that a randomly selected student earns a score between 33 and 48 is 0.3761

Step-by-step explanation:

We compute the z-score related to some value x as [tex]z=\frac{x-50}{10}[/tex]. The z-score related to 33 is given by [tex]z_{1}=\frac{33-50}{10}=-1.7[/tex] and the z-score related to 48 is given by [tex]z_{2}=\frac{48-50}{10}=-0.2[/tex]. Therefore, the probability that a randomly selected student earns a score between 33 and 48 is given by P(-1.7 < Z < -0.2) = P(Z < -0.2) - P(Z < -1.7) = 0.4207 - 0.0446 = 0.3761.

The probability is 0.3761 (a) that a student scores between 33 and 48 on the 4th grade Achievement Test.

To find the probability that a randomly selected student earns a score between 33 and 48 on the 4th grade Achievement Test, we need to use the properties of the normal distribution.

**Standardize the Scores**:

  First, we need to standardize the scores using the formula for z-score:

    [tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]

  where:

  - ( x ) is the score we want to find the probability for,

  - [tex]\( \mu \)[/tex] is the mean (50 in this case),

  - [tex]\( \sigma \)[/tex] is the standard deviation (10 in this case).

  For the lower score [tex]\( x = 33 \)[/tex]:

  [tex]\[ z_{\text{lower}} = \frac{33 - 50}{10} = -1.7 \][/tex]

  For the upper score [tex]\( x = 48 \)[/tex]:

  [tex]\[ z_{\text{upper}} = \frac{48 - 50}{10} = -0.2 \][/tex]

**Find the Probability**:

  Once we have the standardized scores, we can use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores.

  From the z-table or calculator, we find:

  - For [tex]\( z_{\text{lower}} = -1.7 \)[/tex], the corresponding probability is approximately 0.0446.

  - For [tex]\( z_{\text{upper}} = -0.2 \)[/tex], the corresponding probability is approximately 0.4207.

**Calculate the Probability Between the Scores**:

  To find the probability between the two scores, we subtract the probability corresponding to the lower score from the probability corresponding to the upper score:

   [tex]\[ \text{Probability} = \text{Probability}(x < 48) - \text{Probability}(x < 33) \][/tex]

   [tex]\[ \text{Probability} = 0.4207 - 0.0446 \][/tex]

   [tex]\[ \text{Probability} = 0.3761 \][/tex]

So, the correct option is: a) .3761

Complete Question :

A particular type of 4th grade Achievement Test provides overall scores that are normally distributed with a mean of 50 and a standard deviation of 10.                                                                                                           What is the probability that a randomly selected student earns a score between 33 and 48?

a) .3761

b) .4207

c) .4653

d) .0446

Answer with Step-by-step explanation:

We are given  that

Radius of circular disk=r=25 cm

Maximal error in measurement of radius =[tex]\Delta r=[/tex]0.3

We know that

Surface area of circle =A=[tex]\pi r^2[/tex]

Differentiate w.r.t r

a.Maximum error=[tex]dA=2\pi rdr[/tex]

Substitute the values then we get

[tex]dA=2\times 3.14\times 25\times 0.3=47.1 cm^2[/tex]

Hence, the maximum error in area of disk=[tex]47.1 cm^2[/tex]

b.

Relative error in A: [tex]\frac{\Delta A}{A}=\frac{2\pi r\Delta r}{\pi r^2}[/tex]

Substitute the value in the formula

Then ,we get

Relative maximal error in area of disk=[tex]\frac{2\cdot 25}{(25)^2}(0.3)=0.024[/tex]

Hence, the relative maximum error in area  of disk=0.024

c.Relative error  percentage in area of disk=[tex]\frac{\Delta A}{A}\times 100[/tex]

Substitute the values then we get

Relative error  percentage in area of disk=[tex]0.024\times 100=2.4[/tex]%

Hence, the percentage error in area of disk=2.4%

To estimate the maximum error in the calculated area of a disk, use differentials. The maximum error in the area is approximately 15π cm². The relative maximum error is 0.24 and the percentage error is 24%.

To estimate the maximum error in the calculated area of the disk, we can use differentials. The formula for the area of a disk is A = πr², where r is the radius. Taking the differential of this equation gives dA = 2πr dr. Since the maximum error in the radius is 0.3 cm, we substitute dr = 0.3 cm into the equation to find the maximum error in the area. Thus, the maximum error in the calculated area of the disk is approximately 2π(25 cm)(0.3 cm) = 15π cm².

The relative maximum error is the ratio of the maximum error in the area to the actual area. Therefore, the relative maximum error can be calculated as (15π cm²) / (π(25 cm)²) = 15 / 25² = 0.24.

To calculate the percentage error, we multiply the relative maximum error by 100%.

Thus, the percentage error is 0.24 * 100% = 24%.

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Answer:

n=23

Step-by-step explanation:

Answer  explained in the attachment .

Answer:

The answer to your question is b = 148.5

Step-by-step explanation:

To solve this problem, use trigonometric functions

tan Ф = [tex]\frac{opposite side}{adjacent side}[/tex]

opposite side = b

adjacent side = 60

tan Ф = tan 68° = 2.48

Opposite side = adjacent side x tan Ф

Opposite side = 60 x 2.48

Opposite side = b = 148.5

Answer: 24.24 (24 to the nearest whole number)

Step-by-step explanation:

Following trigonometric rule,

Tan tita = opposite side/ adjacent side

Tan 68° = 60 / b

2.475 = 60 / b

By cross multiplication,

2.475b = 60

Divide both sides by 2.475

2.475b/2.475 =60/2.475

b = 24.24

Hence, adjacent side = 24.24 (24 to the nearest whole number since opposite side is in whole number).

Approximately 81.5% of the daily light bulb replacement requests in the state university's physical plant can be expected to number between 63 and 75.

The question is asking for the percentage of daily light bulb replacement requests numbering between 63 and 75 at a state university's physical plant. This can be answered using the 68-95-99.7 rule, also known as the Empirical Rule, which applies to a bell-shaped distribution.

Firstly, identify that a request count of 75 is two standard deviations above the mean (63 + 6*2 = 75), because the standard deviation here is 6. According to the Empirical Rule, approximately 95% of the data lies within two standard deviations of the mean.

However, we are only interested until the upper limit of 75 and not the lower limit (63 - 6*2 = 51). Therefore, we consider half of this 95%, which gives us 47.5%. However, don't forget to include the 34% that lies within the first standard deviation (from 63 to 69). Therefore, approximately 81.5% of the light bulb replacement requests should number between 63 and 75.

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the approximate percentage of light bulb replacement requests numbering between 63 and 75 is 47.5%.

Using the Empirical Rule:

Answer:

  127.3 ft

Step-by-step explanation:

The diagonal of a square is √2 times the length of one side. The distance from home plate to second base is ...

  (90 ft)√2 ≈ 127.3 ft

The catcher at home has to throw the ball 127.3 ft to reach second base.

Answer:

6000π m²/h

Step-by-step explanation:

Area of the circular cloud =πr²----------------------------------------- (1)

Radius as a function of time = R(t)=50+20t------------------------ (2)

dA/dt = (dA/dr) x (dr/dt)-----------------------------------------------------(3)

dA/dr  =  2πr

dr/dt   = 20

After 5 hours, the radius of the cloud will be :

R(5)=50+20(5)

     =   150 meters

Substituting into (3)

dA/dt =   2π(150)  x 20

         =   6000π m²/h

Answer: The correct option is C

Step-by-step explanation:

You begin a job with an annual salary of $32,900

Each year you are assured of a 5.5%. If you get the same amount each year, that is a 100% payment. It is neither reducing nor increasing. But with an increase of 5.5% each year, it means you are getting (100+5.5)% each year. This equals 105.5%. So for each year, you get 105.5% of the previous year.

This is a geometric progression. To determine the total amount that you can earn in 15 years, we will find the sum of 15 terms, S15 of the series. The formula for sum of the nth term of a geometric progression is expressed as

Sn = [a(r^n - 1)] / r - 1

Where

Sn = sum of the nth terms of the series

a = the first term(salary of the first year

r = common ratio

n = number of years.

From the question,

a = 32900

n = 15

r = 105.5/100 = 1.055

S15 = [32900(1.055^15 - 1)] / 1.055 - 1

S15 = [32900(2.23247649224 - 1)] / 0.055

S15 = [32900 × 1.23247649224] / 0.055

S15 = $737245

Answer:

lyrics d  - 1.782

Step-by-step explanation:

Assume Normal Distribution

we have  μ₀ = 40       (from Dullco claims)

And from sample   μ  = 37.8    and standard deviation of  5.4

random sample n = 18

We have to use t-student for testing the hypothesis

and we have a one tail test (left) since Dullco claims : " at least"  meaning (always bigger or at least ) not different.

Then

t (s) =( μ -  μ₀) / (σ/√n)   ⇒ t (s) = [(37.8  - 40 )* √18 ]/5.4

t (s) = - 1.7284

Using the t-distribution, as we have the standard deviation for the sample, it is found that the test statistic is given by:

d. -1.728

At the null hypothesis, it is tested if the batteries last at least 40 hours, that is:

[tex]H_0: \mu \geq 40[/tex]

At the alternative hypothesis, it is tested if they last less than 40 hours, that is:

[tex]H_1: \mu < 40[/tex]

The test statistic is given by:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

The parameters are:

In this problem, the parameters are given by:

[tex]\overline{x} = 37.8, \mu = 40, s = 5.4, n = 18[/tex]

Hence:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]t = \frac{37.8 - 40}{\frac{5.4}{\sqrt{18}}}[/tex]

[tex]t = -1.728[/tex]

Hence option d is correct.

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Answer:

ok with whuth do you need help with

Answer:

The right answer for question 1 is b)  -14

Step-by-step explanation:

Since it's a decline, the amount will be negative.

your second question is not clear in the image

3.5 per day for 4 days so multiply 3.5 by 4, which equals: $14

so the overall change is -$14.00

Answer:

  4.47 years

Step-by-step explanation:

Fill in the given value of P and solve for t.

  15390 = 19000·0.954^t

  15390/19000 = 0.954^t

Taking logarithms ...

  log(1539/1900) = t·log(0.954)

  t = log(1539/1900)/log(0.954) ≈ 4.47 . . . years

Answer:

A

Step-by-step explanation:

Given:

- The original function is:

                               f(x) = x^3

Find:

How does the graph of g(x) = (x − 1)3 + 5 compare to the original function.

Solution:

- We have a general form of the new function relating to parent function.

                               g(x) = a*( x +/- b )^n + c

Where; a, b and c are constants.

- The constant a magnitude denotes steepness of the graph relative to 1 and the sign of a will determine the mirror image of the graph about line y = 0.

- The constant b magnitude denotes shifts of the graph of every x value sign of b will determine the direction of shifts. + b : shift left , - b shift right.

- The constant c magnitude denotes shifts of the graph of every y value. sign of c will determine the direction of shifts. + c : shift up , - b shift down.

- In our g(x).   a = 1 , b = -1 , c = + 5

Hence, g(x) is shifted 1 unit to the right and 5 units up.

The graph of g(x) = (x - 1)^3 + 5 is shifted 1 unit to the right and 5 units up compared to the parent function f(x) = x^3.

The graph of g(x) = (x - 1)3 + 5 is obtained by shifting the graph of f(x) = x3 one unit to the right and five units up. Shifting a function horizontally is done by replacing x with (x - h), where h is the amount of units you want to shift.

In this case, g(x) = (x - 1)3 + 5 is equivalent to f(x - 1) + 5. So the graph is shifted 1 unit to the right. Shifting a function vertically is done by replacing f(x) with f(x) + k, where k is the amount of units you want to shift.

Since f(x - 1) + 5 is obtained by shifting f(x) one unit to the right and five units up, the correct answer is: g(x) is shifted 1 unit to the right and 5 units up.

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Answer:

$43,200

Step-by-step explanation:

Data provided in the question:

Cost of the asset = $80,000

Useful life of the machine = 5 years

Salvage value at the end of useful life = $20,000

Now,

Using the double declining method of depreciation

Annual depreciation rate = 2 ×  [1 ÷ useful life ]

= 2 × [ 1 ÷ 5 ]

= 2 × 0.2

= 0.4 or 40%

thus,

The depreciation from October 1, 2014 to December 31, 2014

= Annual Depreciation rate × duration × Book value for 2014

= 0.4 × 3 months × $80,000

= 0.4 × 0.25 year × $80,000

= $8,000

Book value for 2015

= Cost - depreciation till December 31, 2014

= $80,000 - $8,000

= $72,000

Therefore,

Depreciation for the year 2015

= Annual Depreciation rate  × Book value for 2015

= 0.4 × $72,000

= $28,800

Therefore,

the  book value of the plant asset on the December 31, 2015

= Book value for 2015 - Depreciation for the year 2015

= $72,000 - $28,800

= $43,200

Answer:

  20%

Step-by-step explanation:

Time and speed are inversely proportional, so if Emily's time was 5/4 of normal, her speed was 4/5 of normal, or 1/5 = 20% below normal.

Answer:

The time it takes for the pendulum to swing from its rightmost position to its leftmost position and back again is 1.25 seconds.

Step-by-step explanation:

Given the equation

[tex]d = 11cos( \frac{8\pi}{5}*t)[/tex]-----------Equation 1

where d, in inches of the pendulum of a grandfather clock from the center.

Comparing with the standard equation of an oscillating pendulum bob.

[tex]d = Acos (wt + \alpha )[/tex] ----------Equation 2

         where ω =  angular velocity

                     t =  time taken

                     α =  The angular displacement when t = 0

Comparing equation 1 and 2,

α = 0

[tex]w =\frac{8\pi }{5}[/tex]

Recall that [tex]w = 2\pi f

Therefore,

[tex]2\pi f = \frac{8\pi }{5} \\\\f = \frac{4}{5}[/tex][/tex]

f = 0.8 Hertz

Recall that [tex]f = \frac{1}{T}[/tex]

[tex]T = \frac{1}{0.8}[/tex]

T = 1.25 seconds

Therefore, the time it takes for the pendulum to swing from its rightmost position to its leftmost position and back again is 1.25 seconds.

Answer:

D. There is a significant difference among the reading speeds somewhere.

Hope this helps!

Answer:

1. Value helps businesses allocate resources for their best and most productive uses.

2. The more Scarce a resource, the more costly it will be.

3. A manufacturer that requires scarce and costly resources is likely to charge more for its products.

Step-by-step explanation:

Given:

Three sentences are given we need to add a word or phrase to complete it

First Sentence:

helps businesses allocate resources for their best and most productive uses.

It the above sentence Value will be the word which will complete the sentence because when a resource is valued for his work he provides his best in his work and can be assured as productive too.

Hence  we can say,

Value helps businesses allocate resources for their best and most productive uses.

Second Sentence:

The more a resource, the more costly it will be.

It the above sentence Scarce will be the word which will complete the sentence because Scarce Resource always considered to be more costly.

Hence we can say;

The more Scarce a resource, the more costly it will be.

Third Sentence.

A manufacturer that requires scarce and costly resources is likely to charge for its products.

It the above sentence more will be the word which will complete the sentence because A manufacture who has Scarce and costly resource is said to have less production and hence its product would be more costly too.

Hence we can say;

A manufacturer that requires scarce and costly resources is likely to charge more for its products.

Answer:  Value, scarce , more

Step-by-step explanation:

those are correct