Rewrite expression (4^-2)^3 using positive exponents

Using the concept of Application of Derivatives(A.O.D),we got 4.79 inches will be the square size for making the volume of candy box maximum.

A candy box is made from a piece of a cardboard that measures 43 × 23 inches.

Let squares of equal size will be cut out of each corner with the measure of x inches.  

Therefore, measures of each side of the candy box will become

Length = (43 - 2x)

Width   = (23 - 2x)

Height  = x

Now we have to calculate the value of x for which volume of the box should be maximum.

Volume (V) = Length×Width×Height

 =>V = (43 -2x)×(23 - 2x)×(x)

 =>V= [(43)×(23) - 46x - 86x + 4x²]×x

 =>V= [989 - 132x + 4x²]×x

 =>V= 4x³- 132x² + 989x

Now we find the derivative of V and equate it to 0

[tex]\frac{dV}{dx}[/tex]= [tex]12x^{2} -264x+989[/tex]=0

Now we get values of x by quadratic formula

x=(264±[tex]\sqrt{264^{2}-4.12.989 }[/tex] )/24

=>x=(264±[tex]\sqrt{69696-47472}[/tex])/24

=>x=(264+√22224)/24, and x=(264-√22224)/24

=>x=(264+149.07)/24 and x=(264-149.07)/24

=>x=17.212                  and x=4.79

Now we test it by second derivative test for the maximum volume.

[tex]\frac{d^{n}V }{dx}[/tex]=24x-264

For x = 17.212

[tex]\frac{d^{n}V }{dx}[/tex]=(24×17.212)-264=413.088-264=149.088

This value is > 0 so volume will be minimum.

For x = 4.79

[tex]\frac{d^{n}V }{dx}[/tex]=(24×4.79)-264= -149.04

-149.04 < 0, so volume of the box will be maximum.

Therefore, for x = 4.79 inches volume of the box will be maximum.

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(Complete Question) is

A candy box is made from a piece of cardboard that measures 43 by 23 inches. Squares of equal size will be cut out from  each corner. The sides will then be folded up so that to form a rectangular box. What size square should be cut from each corner so that to obtain maximum​ volume?

If Katerina completes a 15-mile run in 2 1/2 hours. She will complete a mile in an average of five minutes. Option B is correct.

It is described as a single number that indicates either the closed value for each entry in the set of data or the mean value for the entire set of data.

It is given that, In 2 1/2 hours, Katerina completes a 15-mile run.

We have to find the average number of minutes it takes her to run 1 mile.

15 miles = 2 1/2 hours

15 miles=5/2 hours

The unitary approach allows us to calculate the value of a single unit and then use that value to calculate the value of the required number of units.

1 miles= 5 /30 hours

1 miles = 1/6 hours

As we know that 1 hour consists of 60 minutes.

1 hours = 60 minute

1 mile = 60 / 6 minutes

1 miles = 10 minute

Thus, the average number of minutes it takes her to run 1 mile will be 10 minutes. Option B is correct.

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No, Hannah is incorrect.

We are given the expression 42*19.

Hannah got the solution by multiplying 40*10 and 2*9 and then adding the values.

This can be done by using distributive property, which is given by:-

(a + b)(c + d) = ac + ad + bc + bd

We can write,

42*19 = (40 + 2)(10 + 9) = 40*10 + 40*9 + 2*10 + 2*9

Hence, the answer is not just 40*10 + 2*9 but 40*10 + 40*9 + 2*10 + 2*9.

Hannah did not add the remaining 40*9 + 2*10 to calculate the answer.

That is why Hannah was incorrect.

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Step-by-step explanation:

even though the line segment pieces of the horizontal and the inclined lines are of different lengths, but the ratio between the pieces of the same line must be the same.

in other words

x/27 = (32-18)/18 = 14/18 = 7/9

9x/27 = 7

x/3 = 7

x = 7×3 = 21

Answer:0.0008994

Step-by-step explanation:

In the given figure : lines l and m are parallel and a, b are transversal

Since, the lines a and b are parallel and l act as a transversal.

Then,

x = 23

Answer : x = 23

Answer:

Step-by-step explanation:

y = 0.81x + 6.9

residual value = Measured value - Predicted value

Measured value = actual y-coordinate of the point, y

Predicted value = value of y from the equation, y1

residual value = (actual y-coordinate of the point, y) - (value of y from the equation, y1)

residual value = y - y1

x y        y1                residual (y-y1)

3 9     9.33                    -0.33

6 9     11.76                    -2.76

5 13     10.95                   2.05

7 13    12.57                    0.43

8 16     13.38                    2.62

8 11     13.38                   - 2.38

Sum of residuals:

sum = (-0.33) +(-2.76)+(2.05)+(0.43)+(2.62)+(-2.38)

B. −0.37; This indicates that the line of best fit is accurate and is an overall good model for prediction.

(Though not very accurate as it should have been if the sum of residuals was equal to 0).

A total discrepancy of -0.37 is not too bad.

Answer:Answer:

B. −0.37;

This indicates that the line of best fit is accurate and is an overall good model for prediction.

Step-by-step explanation:

y = 0.81x + 6.9

residual value = Measured value - Predicted value

Measured value = actual y-coordinate of the point, y

Predicted value = value of y from the equation, y1

residual value = (actual y-coordinate of the point, y) - (value of y from the equation, y1)

residual value = y - y1

x y        y1                residual (y-y1)

3 9     9.33                    -0.33

6 9     11.76                    -2.76

5 13     10.95                   2.05

7 13    12.57                    0.43

8 16     13.38                    2.62

8 11     13.38                   - 2.38

Sum of residuals:

sum = (-0.33) +(-2.76)+(2.05)+(0.43)+(2.62)+(-2.38)

sum of residuals = -0.37

ANSWER:

B. −0.37; This indicates that the line of best fit is accurate and is an overall good model for prediction.

(Though not very accurate as it should have been if the sum of residuals was equal to 0).

A total discrepancy of -0.37 is not too bad.

Answer:

Step-by-step explanation:

You want the range of amounts of 60% solution that can be added to a 40% solution to achieve 30 kg of a solution that lies in the range of 46% to 50%.

Let x represent the mass in kg of 60% solution added. The fraction of the total solution that is glucose will be ...

  0.46 ≤ (0.60x +0.40(30 -x))/30 ≤ 0.50

Multiplying by 30 and simplifying the inequality, we have ...

  13.8 ≤ 0.20x +12 ≤ 15

  1.8 ≤ 0.2x ≤ 3 . . . . . . . . subtract 12

  9 ≤ x ≤ 15 . . . . . . . . . . divide by 0.2

The least amount of 60% solution that should be added is 9 kg. The greatest amount is 15 kg.

__

Check

  9 kg of 60% +21 kg of 40% has 5.4 +8.4 = 13.8 kg of glucose, the minimum.

  15 kg of 60% +15 kg of 40% has 9 +6 = 15 kg of glucose, the maximum.

(The minimum and maximum values are seen in the solutions steps after the initial multiplication by 30.)

The distance between Ashely’s house and the public library is of 7.8 miles.

Suppose that we have two points with coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The shortest distance between these two points is given by the following rule:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

This formula is derived from the Pythagorean Theorem, as the points form a right triangle in the Cartesian plane, with the hypotenuse representing the distance between them.

In the context of this problem, the coordinates of her house and of the library are given as follows:

Hence the distance is calculated as follows:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

[tex]D = \sqrt{(10 - 4)^2+(7 - 2)^2}[/tex]

D = sqrt(61)

D = 7.8 miles, as each unit on the plane represents one mile.

The problem states that each unit on the plane represents one mile and the image gives their position on the plane.

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The answer of the linear equation for Y=2x+5 is x = (y-5)/2.

What is linear equation?

An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation.  The standard form of a linear equation in one variable is of the form Ax + B = 0.  Now in this equation, x is a variable, A is a coefficient, and B is constant.

The given linear equation is,

Y=2x+5

Now, solve this linear equation for x

Take 5 to LHS it will change sign to -ve

y-5 = 2x

Next, take coefficient of x below to LHS

(y-5)/2 = x

Now, swap the sides

x = (y-5)/2

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Explanation

we have a secant and a tangent

To solve the question, we will apply the intersecting secant and tangent theorem

So, for the question

we have

Therefore, the value of PB is 12

The standard form of a hyperbola with center (h,k) is

We know the vertices are at (5,0) and (-5,0)

The vertices are at (h+a,k) and ( h-a,k)

The center must be at (0,0) and the a value is 5

The coordinates of the foci are at ( h+c,k) and ( h-c, k) and we know one of the foci is at (6,0)

c is equal to 6

We know that c^2 = a^2+b^2

6^2 = 5^2 + b^2

36 = 25+b^2

11 = b^2

We can write the equation

Given data:

Total packs 3x3 sticky notes: 24 packs

Cost of 24 packs: 28.99

Number of sticky notes in a pack: 100

With the information above, we can proceed to find the answers to the questions

Part A: What is the price for each pack?

Since 24 packs cost 28.99

Then 1 pack will cost =

Hence, a pack will cost $1.21

Part B: How much is every single sticky note?

Since each pack cost 1.21, and each pack has 100 sticky notes, then

each sticky note will cost

=> every single sticky note will cost $0.012

Answer:

Explanation:

Here, we want to write the equation of a line in point-slope form

Mathematically, we have that as:

where m, which is the slope is -2/3 and (x1,y1) which is the point is (-3,2)

Substituting the values, we have it that:

Answer:

C

Step-by-step explanation:

1 we need you to provide the function for people to better understand how to answer.

2 Option c was the only one that made sense.

pls mark brainliest

Answer:

The value of k is;

Explanation:

Given the equation:

To solve, let us subtract 8 from both sides;

then, we can then divide both sides by -3;

To solve further we need to take the logarithm of both sides;

finding the value of the log;

solving for k;

Therefore, the value of k is;

The knowledge of place value helped in solving 273÷13 with a rectangular model developed by Carter

The part labelled A is 26 tens

The part labelled B is 13 units

The part labelled C is 13 ones

information given in the question

A rectangle is divided into two sections by a vertical line

The left section is labeled inside with 260

The left edge of the rectangle is labeled 13

other information is in the attached diagram

How to used the model

Carter's model is a rectangle of length 273 and breadth 13.

273 / 13 = 21

This means that dividing the length into 13 places each unit is 13

273 - 13 = 260

hence the 260 represents 20 units, since one unit is subtracted

The knowledge of place value help to get the numbers to be arranged in tens and ones

putting the numbers in tens is done by dividing by 10

= 260 / 10

= 26

hence, A in tens is equivalent to 26

putting the numbers in ones is dine by dividing by 1

C = 13 / 1

C = 13 ones

a unit is 13, B = 13

B covers same space as C so they are equal

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The standard form of the equation of f(x) is option a -x² - 6x + 7

Given,

The graph shows a downward open parabola.

The vertex points are:

(-3, 16), (-7, 0), (1, 0), (-6, 7), (0, 7)

Now,

We have to find the standard form of the function f(x):

As from the graph:

Parabola opens downward, so the function will be negative.

We have the options with:

a = -1, b = -6 and c = 7

Now,

Use x = -b/2a

x = 6/2 × -1 = 6/-2 = -3

Now,

f(x) = -x² - 6x + 7

f(-3) = -(-3)² - 6(-3) + 7

f(-3) = -9 + 18 + 7

f(-3) = 9 + 7

f(-3) = 16

That is, the points for the vertex in this equation is (-3, 16)

Then, the standard form of the equation of f(x) is option a -x² - 6x + 7

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Part (a) Using only the data from the first and last years, build a linear model to describe the cost of individual health insurance from 1999 onward. Use t to represent years after 1999 (treating 1999 as year 0).

we have the ordered pairs

(1999, 2,196) -------> (0,2,196)

(2019,7,186) -------> (20,7,196)

Find out the slope

where

t -----> is the number of years since 1999

P ----> the cost

m=(7,196-2,196)/(20-0)

m=5,000/20

m=250

Find the equation of the linear model in slope-intercept form

P=mt+b

we have

m=250

point (0,2,196)

substitute and solve for b

2,196=250(0)+b

b=2,196

therefore

P=250t+2,196

Part b

Using this linear model, predict the cost of insurance in 2030

For t=2030=2030-1999=31 years

substitute

P=250(31)+2,196

P=$9,946

Part c

According to this model, when do you expect the cost of individual insurance to reach $12,000? Give your answer as a calendar year (ex: 2020).

For P=$12,000

substitute in the linear model

12,000=250t+2,196

250t=12,000-2,196

250t=9,804

t=39 years

therefore

1999+39=year 2038

Part d

Using a calculator or spreadsheet program, build a linear regression model to describe the cost of individual insurance from 1999 onward. Use t to represent years after 1999 (treating 1999 as year 0), and round the values of

P0 and d to the nearest dollar

using a regression calculator

the equation is

ŷ = 239.15065X + 2406.39827

y=239x+2,406

Part e

Using the regression model, predict the cost of insurance in 2030

For t=2030-1999=31 years

P=239(31)+2,406

P=$9,815

Part f

According to the regression model, when do you expect the cost of individual insurance to reach $12,000? Give your answer as a calendar year (ex: 2020).

For P=$12,000

substitute

12,000=239x+2,406

239x=12,000-2,406

239x=9,594

t=40 years

year=1999+40=2039

The height of the box is decreasing at a rate of 2/45 cm/sec.

The volume of a box remains constant at 1100m³ but the area of the base is increasing at a rate of 10 m²/sec.

Since the base of the box is a square, let the sides of the box be a, a, and h. The area of the base can be written as,

A = a²

Differentiate the above equation with respect to t.

dA/dt = 2a(da/dt)

Substitute 10 for dA/dt and 15 for a, to find the rate of change of side of base.

10 = 2(15)(da/dt)

da/dt = 1/3

The volume of the box can be written as,

V = (a)(a)(h) = a²h

Differentiate the above equation with respect to t.

0 = 2a(da/dt) + a² (dh/dt)

dh/dt = -2/a (da/dt)

Substitute 15 for a and 1/3 for da/dt in the above equation, to find the rate of change of height of the box.

dh/dt = -2/15 (1/3)

         = -2/45

Thus, the height of the box is decreasing at a rate of 2/45 cm/sec.

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There are 52 cards in a deck.

There are 12 face cards in the deck, that is 3 face cards per suit.

The probability of getting at least from two draws is given to be:

Therefore, the probability is calculated using the formula:

Using the combination formula, we have the solution to be:

The LAST OPTION is correct.

Answer:

Part A:

Part B:

Step-by-step explanation:

Part A:

Remember that two parallel lines have the same slope. This way, we can conclude that the slope of this particular line is:

Since we already know that this line passes through point (2,6), we can use this point, the slope we've found and the slope-point form to get an equation for the line:

Therefore, we can conclude that the equation of this line is:

Part B:

Remember that the product between the slopes of two perpendicular lines is -1. This way, we'll have that:

Since we already know that this line passes through point (2,6), we can use this point, the slope we've found and the slope-point form to get an equation for the line:

Therefore, we can conclude that the equation of this line is:

We need to solve the following equation for x:

To isolate x, we divide both sides

Answer:

7.3units

Step-by-step explanation:

[tex] \sqrt{(x2 - x1) {}^{2} + (y2 -y1) {}^{2} } [/tex]

[tex] \sqrt{ (- 1 - 6) { \\ }^{2} + (14 - 16) {}^{2} } [/tex]

[tex] = 7.3[/tex]

Answer:

see explanation

Step-by-step explanation:

(a)

given y is inversely proportional to x³ then the equation relating them is

y = [tex]\frac{k}{x^3}[/tex] ← k is the constant of proportion

to find k use any of the ordered pairs from the table

using (1, 8 ) , then

8 = [tex]\frac{k}{1^3}[/tex] = [tex]\frac{k}{1}[/tex] ⇒ k = 8

y = [tex]\frac{8}{x^3}[/tex] ← equation of proportion

(b)

when y = 64 , then

64 = [tex]\frac{8}{x^3}[/tex] ( multiply both sides by x³ )

64x³ = 8 ( divide both sides by 64 )

x³ = [tex]\frac{8}{64}[/tex] = [tex]\frac{1}{8}[/tex] ( take cube root of both sides )

x = [tex]\sqrt[3]{\frac{1}{8} }[/tex] = [tex]\frac{1}{2}[/tex]

The statements The range of the distribution is around 60 cm and

the center of the distribution is around 180 cm are true.

Statistics is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions from data.

In the given statements

There are no outliers in this distribution.

The shape of the boxplot is fairly symmetric.

The range of the distribution is around 60 cm.

The center of the distribution is around 180 cm.

The statement "The range of the distribution is around 60 cm" is true.

The range is the spread of your data from the lowest to the highest value in the distribution.

In the given graph

Range=200-140

=60

So it is true.

The center of the distribution is around 180 cm is also true.

Hence the statements The range of the distribution is around 60 cm and

the center of the distribution is around 180 cm are true.

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Calculating the likelihood of experiments happening is one of the branches of mathematics known as probability.

The probability of getting a king card is 1/13 or 0.077.

A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

Given: Total number of probability - 52

Now, the first selected card is king and it is returned to the deck.

So it contains no effect on the second selection card.

Then, we have to estimate probability to obtain a king

probability = favourable outcomes for a king / total possible outcomess

= 4 king cards / 52 cards in total

= 4 / 52 = 1 / 13

The probability of getting a king card is 1 / 13 or 0.077

Therefore, the correct answer is option B. 1/13, or 0.077.

The complete question is:

The first card selected from a standard 52-card deck was a king. It isreturned to the deck, what is the probability that a king will be drawn on the second selection?

A. 1/4 or 0.25

B. 1/13, or 0.077

C. 12/13, or 0.923

D. 1/3 or 0.33

E. None of the above​

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Answer: 6

Step-by-step explanation:

m = -4 / 2 = -2 / 1 = -2

The transformation from the graph of B to the graph of P is a vertical stretch by a scale factor of 2

From the question, the function definitions are given as

B(m) = 5m

P(m) = 10m

Rewrite the function P(m) as follows

P(m) = 2 x 5m

Substitute B(m) = 5m in the equation P(m) = 2 x 5m

P(m) = 2 x B(m)

When a function is represented as

f(x) = kg(x), then the transformation is a vertical stretch by k

Hence. the transformation a vertical stretch by 2

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This is Geometric progression G P

The next three terms are;

64,128,256