What are the odds against choosing a white or red marble from a bag that contains two blue marbles, one green marble, seven white marbles, and four red marbles?

3:11

3:14

11:3

14:3

(n+4) +7 remove the parenthesis

n+4+7 add the same number answer is n +11

Answer:

12

Step-by-step explanation:

The greatest common divisor(gcd) is also known by the name highest common factor(hcf), greatest common factor(gcf).

Greatest common factor of two number can be defined as the highest  integer that divides both the number.

We have to find greatest common divisor of 252 and 60.

The prime factorization of 252 is:

252 = 2×2×3×3×7

The prime factorization of 60 is:.

60 = 2×2×3×5

Common factors are: 2×2×3

Hence, greatest common divisor of 252 and 60 = 2×2×3 = 12

Answer: 10gtt/ml means that in 10 drops there is a ml of the solution.

Now, you need 1000ml in 8 hours, and want to know the correct rate of flow in drops per minute.

first, 8 hours are 8*60 = 480 minutes.

then you need to infuse 1000ml in 480 minutes, so if you infuse at a constant rate, you need to infuse 1000/480 = 2.083 ml/min.

And we know that 10 drops are equivalent to 1 ml, then 2.083*10= 20.8 drops are equivalent a 2.083 ml, rounding it up, you get 21 drops for the dose.

So the correct rate of flow will be 21 drops per minute.

To find the correct rate of flow for an IV infusion, convert the time to minutes, divide the total volume by the total time to find the rate in ml/min, then multiply by the drip factor to convert to drops/min. Rounding to the nearest drop, we get 21 gtt/min.

To calculate the correct rate of flow for an IV infusion, we need to use the given information: the volume of the IV infusion (D5W 1000 ml), the time over which it must infuse (8 hours), and the IV tubing drip factor (10 gtt/ml).

First, convert the time from hours to minutes as we're interested in drops per minute: 8 hours * 60 minutes/hour = 480 minutes.

Next, we divide the total volume by the total time: 1000 ml / 480 minutes = ~2.08 ml/min. This is the rate in ml/min.

Finally, we multiply by the drip factor to get the rate in drops per minute: 2.08 ml/min * 10 gtt/ml = 20.8 gtt/min.

Rounding to the nearest drop gives us a rate of 21 gtt/min.

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Answer: P means "This statement is false"

then, P is a "function" of some statement,

if i write P( 3> 1932) this could be read as:

3>1932, this statement is false.

You could see that 3> 1932 is false, so P( 3>1932) is true.

Then you could se P(x) at something that is false if x is true, and true if x is false, so p is a negation.

Answer:

[tex]q=9\,,\,r=6[/tex]

Step-by-step explanation:

Division Algorithm :

As per division algorithm , for numbers a and b , there exist numbers q and r such that [tex]a=bq+r\,\,,0\leq r< b[/tex]

Here ,

a = Dividend

b = Divisor

q = quotient

r = remainder

Given : 105 = 11q + r such that [tex]0 \leq r < 11[/tex]

Here, clearly a = 105 , b = 11

To find : q and r

Solution : On dividing 105 by 11 , we get [tex]105=11\times 9+6[/tex]

On comparing [tex]105=11\times 9+6[/tex] with [tex]a=bq+r\,\,,0\leq r< b[/tex] , we get [tex]q=9\,,\,r=6[/tex]

Answer:

You inject 60.9628 milliliters of dosage

Step-by-step explanation:

1 pound = 0.453592kg,

Patient's weight in pounds = 168

Patient's weight in kg = [tex]76.2035  kg[/tex]

Now we are given that The dosage is 0.4 mg per kilogram of bod

So, dosage = [tex]0.4 \times 76.2035 mg = 30.4814 mg[/tex]

1 microgram = 0.001 mg

Concentration of drug = [tex]500 micrograms/ml = 500 * 0.001 mg/ml = 0.5 mg/ml[/tex]

Now we are supposed to find How many milliliters (= cc) are you to inject?

So,milliliters of dosage required to inject = [tex]\frac{30.4814}{0.5} = 60.9628[/tex]

Hence you inject 60.9628 milliliters of dosage

Answer:

The expectation is -$0.189.

Step-by-step explanation:

Consider the provided information.

In European roulette, the wheel is divided into 37 compartments numbered 1 through 36 and 0.

One-half of the numbers 1 through 36 are red, the other half are black, and the number 0 is green.

We need to find the expected value of the winnings on a $7 bet placed on black in European roulette.

Here the half of 36 is 18.

That means 18 compartments are red and 18 are black.

The probability of getting black in European roulette is 18/37

The probability of not getting black in European roulette is 19/37. Because 18 are red and 1 is green.

If the ball lands on a black number, the player wins the amount of his bet.

The bet is ball will land on a black number.

The favorable outcomes are 18/37 and unfavorable are 19/37.

Let S be possible numerical outcomes of an experiment and P(S) be the probability.

The expectation can be calculated as:

E(x) = sum of S × P(S)

For[tex] S_1 = 7[/tex]

[tex]P(S_1) = \frac{18}{37}[/tex]

For [tex]S_2 = -7[/tex](negative sign represents the loss)

[tex]P(S_2) = \frac{19}{37}[/tex]

Now, use the above formula.

[tex]E(x) = 7\times \frac{18}{37}-7\times \frac{19}{37}\\E(x) = -0.189[/tex]

Hence, the expectation is -$0.189.

Answer: There are 3.36 ounces of ground almonds in 1 cup.

Step-by-step explanation:

Since we have given that

1 pint = 0.42 pounds

As we know that

1 cup = 0.5 pints

1 pound = 16 ounces

So, We need to find the number of ounces.

As 1 pint = 0.42 pounds = 0.42 × 16 = 6.72 ounces

0.5 pints is given by

[tex]6.72\times 0.5\\\\=3.36\ ounces[/tex]

Hence, there are 3.36 ounces of ground almonds in 1 cup.

To find out how many ounces are in one cup of ground almonds if 1 pint (2 cups) weighs 0.42 pounds, you divide 0.42 pounds by 2 to get the weight per cup, then multiply by 16 to convert pounds to ounces, resulting in 3.36 ounces per cup.

The question involves converting weight measurements from one unit to another, specifically from pounds to ounces. To do this, you use the conversion factor of 16, because there are 16 ounces in 1 pound. Applying this to the recipe question:

1 pint of ground almonds weighs 0.42 pounds. Since 1 pint equals 2 cups, this weight corresponds to 2 cups of ground almonds.

To find out how many ounces 1 cup of ground almonds weighs, first divide the total weight by the number of cups:

0.42 pounds ÷ 2 cups = 0.21 pounds per cup.

Then, convert pounds to ounces:

0.21 pounds × 16 ounces/pound = 3.36 ounces.

So, for the recipe, you should use 3.36 ounces of ground almonds.

Answer:

160 lbs = 72.57kg

Step-by-step explanation:

This can be solved as a rule of three problem.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.

Unit conversion problems, like this one, is an example of a direct relationship between measures.

Each lb has 0.45kg. How many kg are there in 160lbs. So:

1lb - 0.45kg

160 lbs - xkg

[tex]x = 0.45*160[/tex]

[tex]x = 72.57[/tex] kg

160 lbs = 72.57kg

Answer:

Sample space = {(T,T), (T,H), (HT), (HH)}

Step-by-step explanation:

We are given a fair coin which when tossed one times either gives heads(H) or tails(T).

Now, the same coin is tossed two times.

1) All the possible outcomes

Tails followed by tails

Rails followed by heads

Heads followed by a tail

Heads followed by heads

2) Sample space

{(T,T), (T,H), (HT), (HH)}

3) Formula:

[tex]Probability = \displaystyle\frac{\text{Favourable outcome}}{\text{Total number of outcome}}[/tex]

Using the above formula, we can compute the following probabilities.

Probability((T,T)) =[tex]\frac{1}{4}[/tex]

Probability((T,H)) =[tex]\frac{1}{4}[/tex]

Probability((H,T)) =[tex]\frac{1}{4}[/tex]

Probability((H, H)) =[tex]\frac{1}{4}[/tex]

Probability(Atleast one tails) = [tex]\frac{3}{4}[/tex]

Probability(Atleast one heads) = [tex]\frac{3}{4}[/tex]

Probability(Exactly one tails) = [tex]\frac{2}{4}[/tex]

Probability(Exactly one heads) = [tex]\frac{2}{4}[/tex]

The given sequence is convergent with a limit of -3/2, while the series is divergent since its terms do not approach zero

The sequence in question is An = [6*n/(-4*n + 9)]. To find out if this sequence is convergent or divergent, we need to take the limit as n approaches infinity. As n approaches infinity, the 'n' in the numerator and the 'n' in the denominator will dominate, making the sequence asymptotically equivalent to -6/4 = -3/2. Thus, the sequence is convergent, and its limit is -3/2.

On the other hand, the series ∑n=1[infinity]( An ) is the sum of the terms in the sequence. We can see that as n approaches infinity, the terms of this series do not approach zero, which is a necessary condition for a series to be convergent (using the nth term test). Therefore, the series is divergent.

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Answer:  10

Step-by-step explanation:

The formula to find the sample size is given by :-

[tex]n=(\dfrac{z_{\alpha/2}\ \sigma}{E})^2[/tex]

Given : Significance level : [tex]\alpha=1-0.99=0.1[/tex]

Critical z-value=[tex]z_{\alpha/2}=2.576[/tex]

Margin of error : E=5

Standard deviation : [tex]\sigma=6[/tex]

Now, the required sample size will be :_

[tex]n=(\dfrac{(2.576)\ 6}{5})^2=9.55551744\approx10[/tex]

Hence, the final sample required to be of 10 .

Answer:

the length PQ is 9 units,the length QR is 9 units,the length PR is 9.48 units,the triangle is not a right triangle,this is a isosceles triangle

Step-by-step explanation:

Hello, I think I can help you with this

If  you know two points, the distance between then its given by:

[tex]P1(x_{1},y_{1},z_{1} ) \\P2(x_{2},y_{2},z_{2})\\\\d=\sqrt{(x_{2}-x_{1} )^{2} +(y_{2}-y_{1}  )^{2}+(z_{2}-z_{1} )^{2} }[/tex]

Step 1

use the formula to find the length PQ

Let

P1=P=P(2, −3, −4)

P2=Q=Q(8, 0, 2)

[tex]d=\sqrt{(8-2)^{2} +(0-(-3))^{2}+(2-(-4))^{2}} \\ d=\sqrt{(6)^{2} +(3)^{2}+(6 )^{2}}} \\d=\sqrt{36+9+36}\\d=\sqrt{81} \\d=9\\[/tex]

the length PQ is 9 units

Step 2

use the formula to find the length QR

Let

P1=Q=Q(8, 0, 2)

P2=R= R(11, −6, −4)

[tex]d=\sqrt{(11-8)^{2} +(6-0))^{2}+(-4-2 )^{2}}  \\\\\\d=\sqrt{(3)^{2} +(6)^{2}+(-6 )^{2}}} \\d=\sqrt{9+36+36}\\d=\sqrt{81} \\d=9\\[/tex]

the length QR is 9 units

Step 3

use the formula to find the length PR

Let

P1=P(2, −3, −4)

P2=R= R(11, −6, −4)

[tex]d=\sqrt{(11-2)^{2} +(-6-(-3)))^{2}+(-4-4 )^{2}}  \\\\\\d=\sqrt{(9)^{2} +(-6+3)^{2}+(-4-(-4) )^{2}}} \\d=\sqrt{81+9+0}\\d=\sqrt{90} \\d=9.48\\[/tex]

the length PR is 9.48 units

Step 4

is it a right triangle?

you can check this by using:

[tex]side^{2} +side^{2}=hypotenuse ^{2}[/tex]

Let

side 1=side 2= 9

hypotenuse = 9.48

Put the values into the equation

[tex]9^{2} +9^{2} =9.48^{2}\\ 81+81=90\\162=90,false[/tex]

Hence, the triangle is not a right triangle

Step 5

is it an isosceles triangle?

In geometry, an isosceles triangle is a type of triangle that has two sides of equal length.

Now side PQ=QR, so this is a isosceles triangle

Have a great day

Answer:

(A) No solutions regardless of values of a and b.

Step-by-step explanation:

Asumming that the system of equations is [tex]x+2y-3z=a\\ 2x+4y-6z=2a+2[/tex], the corresponding augmented matrix of the system is [tex]\left[\begin{array}{cccc}1&2&-3&a\\2&4&-6&2a+2\end{array}\right][/tex].

If two time the row 1 is subtracted to row 1, we get the following matrix

[tex]\left[\begin{array}{cccc}1&2&-3&a\\0&0&0&2a+2-2a\end{array}\right][/tex].

Then the system has no solutions regardless of values of a and b.

Answer:

2.68

Step-by-step explanation:

We are given that [tex]x_0=7,x_1=7.1,x_2=7.2,x_3=7.3,x_4=7.4,x_5=7.5[/tex]

h=0.1

y'=2x-y

y(7)=0,f(x,y)=2x-y

[tex]x_0=7,y_0=0[/tex]

We have to find the approximate solution to the initial problem at x=7.1

[tex]y_1=y_0+hf(x_0,y_0)[/tex]

Substitute the value then, we get

[tex]y_1=0+(0.1)(2(7)-0)=0+(0.1)(14)=1.4[/tex]

[tex]y_1=1.4[/tex]

[tex]x_1=x_0+h=7+0.1=7.1[/tex]

[tex]y_2=y_1+hf(x_1,y_1)[/tex]

Substitute the values then, we get

[tex]y_2=1.4+(0.1)(2(7.1)-1.4)=1.4+(0.1)(14.2-1.4)=1.4+(0.1)(12.8)=1.4+1.28[/tex]

[tex]y_2=1.4+1.28=2.68[/tex]

Hence, the approximation solution to the initial problem at x=7.1 is =2.68

The baseball team won 100 games and lost 28 games. We found the number of losses by solving the equation formed by the relationship between wins and losses, and the total number of games played.

To solve the problem, let's denote the number of games the baseball team lost as L, and hence, the games they won would be 3L + 16 as per the condition given. Considering that the team played a total of 128 games, the equation representing the total number of games played is:

L + (3L + 16) = 128

Combining like terms, we get:

4L + 16 = 128

Subtracting 16 from both sides, we have:

4L = 112

Dividing both sides by 4 yields:

L = 28

Now that we have the number of losses, we can calculate the number of wins by substituting L back into 3L + 16:

Wins = 3(28) + 16 = 84 + 16 = 100

Therefore, the team won 100 games and lost 28 games in the season.

Answer:

10.022

Step-by-step explanation:

1. 49/9

2. 106/25

3. 10.022

4. When you add two rational numbers, each number can be written as a :

fraction

5. The sum of two fractions can always  be written as a : fraction

6. Therefore, the  sum of two rational numbers will always be : rational

Answer:

There are 100 milligrams of metoclopramide HCl in each milliliter of the prescription

Step-by-step explanation:

When the prescription says Purified water qs ad 100 mL means that if we were to make this, we should add the quantities given and then, fill it up with water until we have 100 mL of solution, being the key words qs ad, meaning sufficient quantity to get the amount of mixture given.

Then, knowing there is 10 grams of metoclopramide HCl per 100 mL of prescription, that means there is (1 gram = 1000 milligrams) 10000 milligrams of metoclopramide HCl per 100 mL of prescription. That is a concentration given in a mass/volume way.

Knowing the concentration, we can calculate it per mL instead of per 100 mL

[tex]Concentration_{metoclopramide HCL}= \frac{10000mg}{100mL} =100 \frac{mg}{mL}[/tex]

Answer:

The cost is $9.70 per kilogram.

Step-by-step explanation:

This can be solved by a rule of three.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too. In this case, the rule of three is a cross multiplication.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.

In this problem, the measures are the weight of the cheese and the price. As the weight increases, so does the price. It means that this is a direct rule of three.

Solution:

The problem states that cheese costs $4.40 per pound. Each kg has 2.2 pounds. How many kg are there in 1 pound. So:

1 pound - xkg

2.2 pound - 1 kg

[tex]2.2x = 1[/tex]

[tex]x = \frac{1}{2.2}[/tex]

[tex]x = 0.45[/tex]kg

Since cheese costs $4.40 per pound, and each pound has 0.45kg, cheese costs $4.40 per 0.45kg. How much does is cost for 1kg?

$4.40 - 0.45kg

$x - 1kg

[tex]0.45x = 4.40[/tex]

[tex]x = \frac{4.40}{0.45}[/tex]

[tex]x = 9.70[/tex]

The cost is $9.70 per kilogram.

"The cost per kilogram of cheese is approximately $2.00.

To find the cost per kilogram, we need to convert the cost from dollars per pound to dollars per kilogram using the conversion factor between pounds and kilograms. Given that 1 kilogram is equal to 2.2 pounds, we can set up the following conversion:

Cost per pound of cheese = $4.40

Conversion factor = 2.2 pounds/kilogram

Now, to find the cost per kilogram, we divide the cost per pound by the conversion factor:

Cost per kilogram = Cost per pound / Conversion factor

Cost per kilogram = $4.40 / 2.2 pounds/kilogram

Performing the division, we get:

Cost per kilogram ≈ $2.00

Answer: 15

Step-by-step explanation:

Given : The number of kinds of ice-cream cones ( sugar cones, cake cones and waffle cones)=3

The number of flavors of ice-creams =5

By using the fundamental principle of counting , we have

The number of possible cone/ice cream combinations can be ordered will be :-

[tex]5\times3=15[/tex]

Hence, the number of possible cone/ice cream combinations can be ordered =15

Answer: There is increase of 4.255 in the amount of carbon in the atmosphere.

Step-by-step explanation:

Since we have given that

Concentration of carbon in Earth's atmosphere in 1800 = 280 ppm

Concentration of carbon in Earth's atmosphere in 2015 = 399 ppm

We need to find the percentage increase in the amount of carbon in the atmosphere.

So, Difference = 399-280 = 119 ppm

so, percentage increase in the amount of carbon is given by

[tex]\dfrac{Difference}{Original}\times 100\\\\=\dfrac{119}{280}\times 100\\\\=\dfrac{11900}{280}\\\\=42.5\%[/tex]

Hence, there is increase of 4.255 in the amount of carbon in the atmosphere.

Answer:

Ans. Joe will have $720,862.28 and Jill will have $819,348.90 after 30 years.

Step-by-step explanation:

Hi, since the interest is compounded with each payment, the effective rate of Joe is exactly equal to its compounded rate, that is 9.3%, but in the case of Jill, this rate is compounded weekly, this means that we have to divide 9.3% by 52 (which are the weeks in a year) in order to obtain an effective rate, in our case, effective weekly.

On the other hand, the time for Joe is pretty straight forward, he saves for 30 years at an effective annual interest rate of 9.3%, but Jill saves for 30*52=1560 weeks, at a rate of 0.1788% effective weekly.

They both have to use the following formula in order to find how much money will they have after 30 years of savings.

[tex]FutureValue=\frac{A((1+r)^{n}-1) }{r}[/tex]

In the case of Joe, this should look like this

[tex]FutureValue=\frac{5,000((1+0.093)^{30}-1) }{0.093} =720,862.28[/tex]

In the case of Jill, this is how this should look like.

[tex]FutureValue=\frac{96.15((1+0.001788)^{1560}-1) }{0.001788} =819,348.90[/tex]

Best of luck.

Answer:

The man had a displacement of 12.88 m southeastward

Step-by-step explanation:

The path of man forms a right triangle. The first two magnitudes given in the problem form the legs and the displacement that we must calculate forms the hypotenuse of the triangle. To do this we will use the equation of the pythagorean theorem.

H = magnitude of displacement

[tex]H^2 = \sqrt{L_1^2 + L_2^2} = \sqrt{6.70^2 + 11.0^2} = \sqrt{165.89}   =12.88 m[/tex]

using the graphic method, we will realize that the displacement is oriented towards the southeast

Step-by-step explanation:

We want to show that

[tex]=A \setminus (B \cup C) = (A\setminus B) \cap (A\setminus C)[/tex]

To prove it we just use the definition of [tex]X\setminus Y = X \cap Y^c[/tex]

So, we start from the left hand side:

[tex]=A \setminus (B \cup C) = A \cap (B \cup C)^c[/tex] (by definition)

[tex]=A \cap (B^c \cap C^c)[/tex] (by DeMorgan's laws)

[tex]=A \cap B^c \cap C^c[/tex] (since intersection is associative)

[tex]=A \cap B^c \cap A \cap C^c[/tex] (since intersecting once or twice A doesn't make any difference)

[tex]=(A \cap B^c) \cap (A \cap C^c)[/tex] (since again intersection is associative)

[tex]=(A\setminus B) \cap (A \setminus C)[/tex] (by definition)

And so we have reached our right hand side.

Answer:

There is a 35.7% probability that this deal will not materialize.

Step-by-step explanation:

This problem can be solved by a simple system of equations.

-I am going to say that x is the probability that this deal materializes and y is the probability that this deal does not materialize.

The sum of all probabilities is always 100%. So

[tex]1) x + y = 100[/tex].

Bob, the proprietor of Midland Lumber, believes that the odds in favor of a business deal going through are 9 to 5.

Mathematically, this means that:

[tex]2) \frac{x}{y} = \frac{9}{5}[/tex]

We want to find the value of y. So, we can write x as a function of y in equation 2), and replace it in equation 1).

Solution:

[tex]\frac{x}{y} = \frac{9}{5}[/tex]

[tex]x = \frac{9y}{5}[/tex]

[tex]x + y = 100[/tex]

[tex]\frac{9y}{5} + y = 100[/tex]

[tex]\frac{14y}{5} = 100[/tex]

[tex]14y = 500[/tex]

[tex]y = \frac{500}{14}[/tex]

[tex]y = 35.7[/tex]

There is a 35.7% probability that this deal will not materialize.

Answer:

66.11

Step-by-step explanation:

We are given that  a number

66.1086

We have to round the number to hundredths

Place of 6=One;s

Place of second 6=Tens

Place of 1=Tenths

Place of 0=Hundredths

Place of 8=Thousandths

Place of 6=Ten thousandths

Thousandths place is 8 which is greater than 5 therefore, one will be added to hundredth place and other number on the left side of  hundredth place remain same and the numbers on the right side of hundredth place will be replace by zero.

Therefore, the given number round to hundredths=66.11

Answer: 0.0011

Step-by-step explanation:

By using the standard normal distribution table , the probability that Z is to the left of 3.05 is [tex]P(z<3.05)= 0.9989[/tex]

We know that the probability that Z is to the right of z is given by :-

[tex]P(Z>z)=1-P(Z<z)[/tex]

Similarly,  the probability that Z is to the right of 3.05 will be :-

[tex]P(Z>3.05)=1-P(Z<3.05)=1-0.9989=0.0011[/tex]

Hence, the probability that Z is to the right of 3.05 = 0.0011

Answer:

a) The set of solutions is [tex]\{(0,-3x_3,x_3): x_3\; \text{es un real}\}[/tex] y b) the set of solutions is [tex]\{(-6,\frac{-41}{17}-\frac{30}{17}x_4 , \frac{37}{17}+\frac{8}{17} x_4 ,x_4): x_4\;\text{es un real}\}[/tex].

Step-by-step explanation:

a) Let's first find the echelon form of the matrix [tex]\left[\begin{array}{ccc}5&2&6\\-2&1&3\end{array}\right][/tex].

The system has a free variable (x3).

       x1+[tex]\frac{2}{5}[/tex](-3x3)+[tex]\frac{6}{5}[/tex]x3=

      x1-[tex]\frac{6}{5}[/tex]x3+[tex]\frac{6}{5}[/tex]x3

     then x1=0.

The system has infinite solutions of the form (x1,x2,x3)=(0,-3x3,x3), where x3 is a real number.

b) Let's first find the echelon form of the aumented matrix [tex]\left[\begin{array}{ccccc}1&-2&1&-4&1\\1&3&7&2&2\\0&-12&-11&-16&5\end{array}\right][/tex].

The system has a free variable (x4).

      x2=[tex]\frac{-41}{17}-\frac{30}{17}[/tex]x4.

The system has infinite solutions of the form (x1,x2,x3,x4)=(-6,[tex]\frac{-41}{17}-\frac{30}{17}[/tex]x4,[tex]\frac{37}{17}[/tex]+ [tex]\frac{8}{17}[/tex]x4,x4), where x4 is a real number.

Consider the cubic polynomial,

[tex](x+a)(x+b)(x+c)[/tex]

Expanding this gives

[tex]x^3+(a+b+c)x^2+(ab+ac+bc)x+abc=x^3+x^2-4x-4[/tex]

We can factor this by grouping,

[tex]x^3+x^2-4x-4=x^2(x+1)-4(x+1)=(x^2-4)(x+1)=(x-2)(x+2)(x+1)[/tex]

Then letting [tex]a=-2[/tex], [tex]b=2[/tex], and [tex]c=1[/tex] gives [tex]a^3+b^3+c^3=-8+8+1=\boxed1[/tex]

Answer:

Ben must make at least 14 deliveries to reach his goal.

Step-by-step explanation:

The problem states that Ben earns $9 per hour and $6 for each delivery he makes. So his daily earnings can be modeled by the following function.

[tex]E(h,d) = 9h + 6d[/tex],

in which h is the number of hours he works and d is the number of deliveries he makes.

He wants to earn more than $155 in an 8 hour work day.What is the least number of deliveries he must make to reach his goal?

This question asks what is the value of d, when E = $156 and h = 8. So:

[tex]E(h,d) = 9h + 6d[/tex]

[tex]156 = 9*8 + 6d[/tex]

[tex]156 = 72 + 6d[/tex]

[tex]6d = 84[/tex]

[tex]d = \frac{84}{6}[/tex]

d = 14

Ben must make at least 14 deliveries to reach his goal.