A sample of iron ore has a density of 7.87 what is its mass? Show all of your work. If it measures 7.5 cm x 2.53 em x 7.15 cm,

Step-by-step explanation:

(a) 4 and 11

   binary equivalent of 4 in 5 bit = 00100

   binary equivalent of 11 in 5 bit = 01011

   decimal number 4 in 2's complement form = 11100

   decimal number 11 in 2's complement form = 10101

now,

        1 1 1 0 0

     + 1 01 0 1

    1 1 000 1

Since, we are doing addition on 5 bit numbers but the result of addition came in 6 digit, so there will be overflow.

(b) 6 and 14

   binary equivalent of 6 in 5 bit = 00110

   binary equivalent of 14 in 5 bit = 01110

   decimal number 6 in 2's complement form = 11010

   decimal number 14 in 2's complement form = 10010

now,

        1 1 0 1 0

     + 1 00 1 0

    1 0 1 1 0 0

Since, we are doing addition on 5 bit numbers but the result of addition came in 6 digit, so there will be overflow.

(c) -13 and 12

   binary equivalent of -13 in 5 bit = 10011

   binary equivalent of 12 in 5 bit = 01100

   decimal number -13 in 2's complement form = 01101

   decimal number 12 in 2's complement form = 10100

now,

        0 1 1 0 1

     + 1 0 1 0 0

    1 0 0 0 0 1

Since, we are doing addition on 5 bit numbers but the result of addition came in 6 digit, so there will be overflow.

(d) -4 and 8

   binary equivalent of -4 in 5 bit = 11100

   binary equivalent of 8 in 5 bit = 01000

   decimal number -4 in 2's complement form = 00100

   decimal number 8 in 2's complement form = 11000

now,

        0 0 1 0 0

     + 1  1 0 0 0

       1  1  1 0 0

Since, we are doing addition on 5 bit numbers and the result of addition also came in 5 digit, so there will  not be overflow.

(e) -2 and -9

   binary equivalent of -2 in 5 bit = 11110

   binary equivalent of -9 in 5 bit = 10111

   decimal number -2 in 2's complement form = 00010

   decimal number -9 in 2's complement form = 01001

now,

        0 0 0 1 0

     + 0  1 0 0 1

       0  1  0 1 1

Since, we are doing addition on 5 bit numbers and the result of addition also came in 5 digit, so there will  not be overflow.

(f) -9 and -14

   binary equivalent of -9 in 5 bit = 10111

   binary equivalent of -14 in 5 bit = 10010

   decimal number -9 in 2's complement form = 01001

   decimal number -10 in 2's complement form = 01110

now,

        0 1 0 0 1

     + 0 1  1 1  1

       1  1  000

Since, we are doing addition on 5 bit numbers and the result of addition also came in 5 digit, so there will  not be overflow.

To convert decimal numbers to 5-bit 2's-complement numbers, convert each number to binary and add them.

(a) For 4, convert to binary: 00100. For 11, convert to binary: 01011. Add the binary numbers: 00100 + 01011 = 01111. Since the sum is positive, there is no overflow.

(b) For 6, convert to binary: 00110. For 14, convert to binary: 01110. Add the binary numbers: 00110 + 01110 = 10100. Since the sum is negative, there is overflow.

(c) For -13, convert to binary: 10011. For 12, convert to binary: 01100. Add the binary numbers: 10011 + 01100 = 11111. Since the sum is negative, there is no overflow.

(d) For -4, convert to binary: 11100. For 8, convert to binary: 01000. Add the binary numbers: 11100 + 01000 = 00100. Since the sum is negative, there is overflow.

(e) For -2, convert to binary: 11110. For -9, convert to binary: 10111. Add the binary numbers: 11110 + 10111 = 101001. Since the sum is positive, there is overflow.

(f) For -9, convert to binary: 10111. For -14, convert to binary: 10010. Add the binary numbers: 10111 + 10010 = 110001. Since the sum is negative, there is overflow.

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Answer:

0 is not an element of an empty set.

Step-by-step explanation:

We are asked to determine whether 0 is not an element of an empty set.

We know that an empty set is an unique set having no elements. The cardinality of an empty set is 0.

Cardinality stands for the count of element is an set. An empty set is denoted by symbols ∅ or { }.

The empty set is like an empty container. The container is there, but nothing is in it.

When 0 is an element of a set, then its cardinality would be 1.

Therefore, 0 is not an element of an empty set.

Answer:

The amount is sufficient to make 75 tarts.

Step-by-step explanation:

We have been given that a recipe for individual chocolate hazelnut tarts calls for ½ cup of hazelnuts per tart and 1 cup of hazelnuts weighs 4 ounces.

The half cup of hazelnuts will weigh 2 ounces [tex](\frac{4}{2}=2)[/tex].

1 kg equals 35.274 ounces.

[tex]\text{5 kg}=5\times 35.274\text{ ounces}[/tex]

[tex]\text{5 kg}=176.37\text{ ounces}[/tex]

Since each tart needs  ½ cup of hazelnuts and half cup of hazelnuts will weigh 2 ounces, so we will divide 176.37 ounces by 2 to find number of tarts.

[tex]\frac{176.37}{2}=88.185\approx 88[/tex]

Since we can make 88 tarts from 5 kg hazelnuts, therefore, the 5-kilogram bag of hazelnuts be sufficient to make 75 tarts.

Answer:

The number of pennies,nickels and dimes are (p,n,d)=(3,4,6).

Further explanation:

Given:

A box holding pennies, nickels and dimes contains thirteen coins in a box.

Total value is 83 cents.

Calculation:

Consider p,n and d be the number of pennies, nickel and dimes.

Now, total is 13 coins so [tex]p+n+d=13[/tex]

As we know that these following are US coin.

Penny=1 cent

Nickel=5 cents

Dime=10 cents

Step 1:

The value is already given as 83 cents that is 80+3 cents.

80 cents can be possible in many combinations as follows:

(N,D)=(0,8),(2,7),(4,6),(6,5),(8,4),(10,3),(12,2),(14,1),(16,0)

It is given that the total number of cents is 13 so we choose (4,6) as (n,d) .

So the value of nickel n=4

Dimes d=6

Step 2:

The value of p is calculated as follows:

Substitute 4 for n, 6 for d in equation [tex]p+n+d=13[/tex] as follows:

[tex]p+4+6=13[/tex]

[tex]p+10=13[/tex]

[tex]p=13-10[/tex]

p=3

Thus, the number of pennies,nickels and dimes are (p,n,d)=(3,4,6).

The problem can be represented by two equations based on the total coins and their total value. The number of pennies, nickels, and dimes cannot be precisely determined without extra constraints or assumptions.

This problem is about the trio of pennies, nickels, and dimes, and their corresponding values; 1 cent, 5 cents, and 10 cents, respectively. Let's denote the number of pennies as P, nickels as N, and dimes as D.

From the problem, we know two key pieces of information:

Mind this information, we'll try to solve it using the system of linear equations method. However, it's impossible to precisely calculate the number of each coin type without additional constraints or assumptions. This math problem is a common example of how real-life situations can pose complex mathematical challenges, requiring more information or advanced techniques of problem solving.

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Answer:

1) $329.66

2) $8011.20

3) 3.2%

4) 4

Step-by-step explanation:

Simple interest formula: I = P*r*t

the simple interest on the loan: $875 at 6.85% for 5 years 6 months

5 years 6 months = 5,5 years

6.85% = 0.0685

I = 875*0.0685*5.5 = 329.66

the total amount due for the simple interest loan: $6400 at 5.3% for 4 years 9 months.

4 years 9 months = 4 + 9/12 = 4 + 0.75 = 4.75

5.3% = 0.053

I = 6400*0.053*4.75 = 1611.20

Total amount due: 6400+1611.20 = 8011.20

the interest rate on a loan charging $960 simple interest on a principal of $3750 after 8 years.

I = 960

P = 3750

t = 8

960 = 3750*r*8

960 = 30000*r

r = 0.032

r = 3.2%

the term of a loan of $350 at 4.5% if the simple interest is $63.

P = 350

r = 4.5% = 0.045

I = 63

t = ?

63 = 350*0.045*t

63 = 15.75*t

t = 4

Answer:

You should take 2 tablets of Aspirin.

Six tablets is not the correct dosage.

You should get 4 tablets.

Step-by-step explanation:

You wake up with a fever! All you can find at the store is Aspirin, and the bottle says to take 162 mg. The bottle also says each tablet has 81 mg in each tablet. How many tablets should you take?

This can be solved by this following rule of three.

1 tablet - 81mg

x tablets - 162mg

[tex]81x = 162[/tex]

[tex]x = \frac{162}{81}[/tex]

[tex]x = 2[/tex]

You should take 2 tablets of Aspirin.

The doctor orders 120 mg of medicine to be given twice a day. The nurse comes in with 6 tablets and tells you that there are 30 mg in each tablet. Is this the correct dosage?

We need to see how many mg are in 6 tablets. If there are 120mg in 6 tablets, this is the correct dosage. We verify this by the following rule of three:

1 tablet - 30mg

6 tablets - x mg

[tex]x = 30*6[/tex]

[tex]x = 180[/tex]mg.

In 6 tablets, there are 180mg. So, it is not the correct dosage.

If not, how many tablets should you get?

Knowing that each tablet has 30mg, in how many tablets are there 120mg?

1 tablet - 30 mg

x tablets - 120 mg

[tex]30x = 120[/tex]

[tex]x = \frac{120}{30}[/tex]

[tex]x = 4[/tex]

You should get 4 tablets.

Answer:

It will take Amy approximately 5.28 hours to count to 19,000.

Step-by-step explanation:

The first step to solve this problem is finding how many minutes it takes for Amy to count to 19,000. In each minute, she counts 60 numbers. So

1 minutes - 60 numbers

x minutes - 19,000 numbers

[tex]60x = 19,000[/tex]

[tex]x = \frac{19,000}{60}[/tex]

[tex]x = 316.7 minutes[/tex]

It is going to take 316.7 minutes for Amy to count to 19,000. How many hours are 316.7 minutes? Each hour has 60 minutes, so:

1 hour - 60 minutes

x hours - 316.7 minutes

[tex]60x = 316.7[/tex]

[tex]x = \frac{316.7}{60}[/tex]

[tex]x = 5.28[/tex]

It will take Amy approximately 5.28 hours to count to 19,000.

Answer:

In order to maximize the profit pairs of pants jackets, the manufactures should make 6 pants and 3 jackets.

Step-by-step explanation:

This problem can be solved by a system of first order equations:

I am going to say that [tex]x[/tex] is the number of pants and [tex]y[/tex] is the number of jackets that the manufacturer should make.

The profit will be maximized when all the avaiable time of the sewing operator and the cutter time has been used.

The problem states that there are 60 minutes of sewing operator time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. So:

[tex]8x + 4y = 60[/tex]

We can simplify this equation by 4. So:

[tex]2x + y = 15[/tex]

The problem also states that there are 48 minutes of cutter time available. Cutters take 4 minutes on pants and 8 minutes on a jacket. So:

[tex]4x + 8y = 48[/tex]

Again simplifying by 4,

[tex]x + 2y = 12[/tex].

Now we have to solve the following system:

[tex]2x + y = 15[/tex]

[tex]x + 2y = 12[/tex]

I am going to write y as a function of x in the first equation, and replace it in the second.

[tex]y = 15 - 2x[/tex]

[tex]x + 2y = 12[/tex]

[tex]x + 2(15 - 2x) = 12[/tex]

[tex]x + 30 - 4x = 12[/tex]

[tex]-3x = -18[/tex]

[tex]3x = 18[/tex]

[tex]x = \frac{18}{3}[/tex]

[tex]x = 6[/tex]

Now, replacing

[tex]y = 15 - 2x = 15 - 2(6) = 15 - 12 = 3[/tex]

In order to maximize the profit pairs of pants jackets, the manufactures should make 6 pants and 3 jackets.

Answer:

The first one: It is in the 500,000 place.

The second one: It is in the 50 place.

Step-by-step explanation:

Answer:

Irrational numbers are not closed under addition.

Step-by-step explanation:

Irrational numbers are the numbers that cannot be expressed in the form of a fraction [tex]\frac{x}{y}[/tex]. In other words we can say that irrational number,s decimal expantion does not cease to end.

The closure property of addition in irrational numbers say that sum of two irrational number is always a rational number, But this is not true. It is not necessary that the sum is always irrational some time it may be rational.

This can be understood with the help of an example:

let (2+√2) and (-√2) be two irrational number. Their sum is (2+√2)+(-√2) = 2, which is clearly a rational number.

Hence, irrational numbers are not closed under addition.

Answer:    [tex]\dfrac{1}{36}[/tex]

Step-by-step explanation:

We know that when two events A and B are independent , then the probability of getting A and B will be :-

[tex]P(A\cap B)=P(A)\times P(B)[/tex]

In a fair dice,total outcomes =6

Number of 2's =1

Then, the probability of getting a 2 =[tex]\dfrac{1}{6}[/tex]

If you roll one die two times, then the probability of getting a 2 on the first roll and a 2 on the second roll will be:-

[tex]\dfrac{1}{6}\times\dfrac{1}{6}=\dfrac{1}{36}[/tex]

Hence, the probability of getting a 2 on the first roll and a 2 on the second roll=[tex]\dfrac{1}{36}[/tex]

Answer:

a) You should invest $6941.90 today.

b) The effective annual interest rate is 11%.

c) t is approximately 6.

Step-by-step explanation:

These are compound interest problems. The compound interest formula is given by:

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

Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

a) How much would you invest today to have $9500 in 8 years if the effective annual rate of interest is 4%?

Here, we want to find the value of P when [tex]A = 9500, t = 8, n = 1, r = 0.04[/tex].

[tex]9500 = P(1 + \frac{0.04}{1})^{8}[/tex]

[tex]P = \frac{9500}{1.3685}[/tex]

[tex]P = 6941.90[/tex].

You should invest $6941.90 today.

b) Suppose that an investment of $5750 accumulates to $11533.20 at the end of 13 years, then the effective annual interest rate is i= ?

Here, we have that [tex]A = 11533.20, P = 5750, t = 13, n = 1[/tex], and we want to find the value of i, that is r on the formula above the solutions.

[tex]11533.20 = 5750(1 + r)^{13}[/tex]

[tex]\sqrt[13]{11533.20} = \sqrt[13]{5750(1 + r)^{13}}[/tex]

[tex]2.05 = 1.94(1 + r)[/tex]

[tex]r = 0.11[/tex]

The effective annual interest rate is 11%.

c) At an effective annual rate of interest of 5.3%, the present value of $7425.70 due in t years is $3250. Determine t.

Here, we have that [tex]A = 7425.70 + 3250 = 10675.7, P = 7425.7, r = 0.053, n = 1[/tex] and we have to find t. So

[tex]10675.7 = 7425.7(1 + \frac{0.053}{1})^{t}[/tex]

[tex](1 + 0.053)^{t} = \frac{10675.7}{7425.7}[/tex]

[tex](1.0553)^{t} = 1.4377[/tex]

We have that:

log_{a}a^{n} = n

So

[tex]log_{1.0553} (1.0553)^{t}  = log_{1.0553} 1.4377[/tex]

[tex]t = 5.95[/tex]

t is approximately 6.

Answer:3 9 15 21

Step-by-step explanation:

3+6=9

9+6=15

15+6=21

or think of it as 3+3+3+3+3+3+3=21

Final answer:

Adam's number pattern starts with 3, and by adding 6 to each previous term, the first four terms are 3, 9, 15, and 21.

Explanation:

The student is asking about creating a number pattern based on a given rule. In this instance, the pattern begins with the number 3, and the rule is to add 6 to the previous term to get the next term. To determine the first four terms of Adam's pattern, we start with the number 3 and repeatedly add 6.

First term: 3 (starting number)

Second term: 3 + 6 = 9

Third term: 9 + 6 = 15

Fourth term: 15 + 6 = 21

Therefore, the first four terms of Adam's number sequence are 3, 9, 15, and 21.

Answer: 18.75%

Step-by-step explanation:

The percent strength of a solution is given by :-

[tex]\% \text{ Strength}=\dfrac{\text{Mass of solute in g}}{\text{Volume of solution in mL}}\times100[/tex]

Given :  A solution of ibuprofen contains 150g in 800mL .

Then, the percent strength of this solution will be :-

[tex]\% \text{ Strength}=\dfrac{150}{800}\times100\\\\\Rightarrow\ \% \text{ Strength}=18.75\%[/tex]

Hence, the percent strength of this solution = 18.75%

The 99% confidence interval for the true mean time to change a water pump is approximately (14.650, 20.850) minutes, based on the recorded times.

To find the 99% confidence interval for the true mean time it takes to change a water pump, follow these steps in Excel:

Compute the sample mean [tex](\(\bar{x}\))[/tex] using the AVERAGE function and the sample standard deviation (s) using the STDEV.S function for the recorded times.

Calculate the degrees of freedom (n - 1) using the COUNT function to count the number of samples and subtracting 1.

Use the T.INV.2T function to find the critical value for the 99% confidence level with the obtained degrees of freedom.

Compute the margin of error using the formula:

[tex]\(t_{\alpha/2} \times \frac{s}{\sqrt{n}}\)[/tex], where [tex]\(t_{\alpha/2}\)[/tex] is the critical value, s is the sample standard deviation, and n is the sample size.

Determine the confidence interval by subtracting and adding the margin of error to the sample mean.

Round the lower and upper bounds of the confidence interval to three decimal places.

Following these steps, the 99% confidence interval for the true mean time to change a water pump is approximately (14.650, 20.850) minutes based on the given recorded times.

Answer: There are 18 sets of five marbles including at least two red ones.

Step-by-step explanation:

Since we have given that

Number of red marbles = 3

Number of green marbles = 3

Number of lavender marbles = 1

total number of marbles = 3+3+1+1 = 8

We need to find the sets of five marbles including at least two red ones.

so, it becomes,

[tex]^3C_2\times ^4C_3+^3C_3\times ^4C_2\\\\=18[/tex]

hence, there are 18 sets of five marbles including at least two red ones.

To determine the total number of sets of five marbles that include at least two red ones, you need to figure out the total number of ways you can choose 5 marbles out of the 8 in the bag, and then calculate combinations for cases with at least 2 red ones and add them up.

To determine the total number of sets of five marbles that include at least two red ones, you need to use combinatorics, which is a branch of Mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those posed by questions concerning the possibility of constructing.

Firstly, you need to figure out the total number of ways you can choose 5 marbles out of the 8 in the bag. This is calculated using the combination formula: C(n, k) = n! / [k!(n-k)!]. Where n is the total number of items, and k is the number of items to choose. For this case, n is 8 (total marbles) and k is 5 (marbles we want to choose), so total possible combinations will be C(8, 5).

We want sets that include at least two red marbles, so the sets can have 2, 3 or all red marbles. So, calculate combinations for each case and add them up, like this:

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Answer: So, if m divides n then n/m = x, and x is integer.

then -n/m = (-1*n)/m = -1*n/m = -1*x = -x.

So if x is integer, -x also is integer, then -n/m is integer and then m divides -n.

Where you used that in the integers set each number a has a opposite (also in the set ) such that a + b= 0, and b = -a = -1*a.

Answer:

The volume of this tank is [tex]V = 10956.30 ft^{3}[/tex], using [tex]\pi = 3.14[/tex]

Step-by-step explanation:

A tank has the format of a cylinder.

The volume of the cylinder is given by:

[tex]V = \pi r^{2}h[/tex]

In which r is the radius and h is the heigth.

The problem states that the diameter is measured carefully to be 15.00 ft. The radius is half the diameter. So, for this tank

[tex]r = \frac{15}{2} = 7.50[/tex] ft

The height of the tank is 62 ft, so [tex]h = 62[/tex].

The volume of this tank is:

[tex]V = \pi r^{2}h[/tex]

[tex]V = pi*(7.5)^2*62[/tex]

[tex]V = 10956.30 ft^{3}[/tex]

The volume of this tank is [tex]V = 10956.30 ft^{3}[/tex], using [tex]\pi = 3.14[/tex]

Answer:

-B9C

Step-by-step explanation:

Hi!

Firstly,

1) Start dividing -2972 : 16 = -185 (quotient) *(16) -12 Remainder

2) Do it again! Divide -185 for 16 = -185 / 16 = -11 (quotient) *(16) - 9 Remainder

3) Divide = -11/16 there's no integer result (since it's 0.68) we put it 0*16 -11 (Remainder) = 11

(Since the result gave us a 0 as integer. We had to lower it one unit the Remainder to satisfy the division algorithm which says = a:b=q*b +r,

11 =0*16+11

4) Gathering all Remainders from bottom to top: 12912

Comparing with the Table (below), from the last remainder to the first, and checking it with the table:

Decimal = Hex (multiplying by minus 1 since it's negative):

-11912 =-B9C

Answer:

The probability that the sample mean would differ from the population mean by greater than 2.2 kilograms is 0.0104 .

Step-by-step explanation:

The mean weight of an adult is 62 kilograms with a variance of 144

i.e. [tex]\mu = 62 \\\sigma^2 = 144[/tex]

We are supposed to find probability that the sample mean would differ from the population mean by greater than 2.2 kilograms

i.e. [tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=1-P(59.8<\bar{x}<64.2)[/tex]

Using formula : [tex]\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

[tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=1-P(59.8<\bar{x}<64.2)[/tex]

[tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=1-P(\frac{59.8-62}{\frac{12}{\sqrt{195}}}<\frac{64.2-62}{\frac{12}{\sqrt{195}}})[/tex]

[tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=1-P(-2.56<z<2.56)[/tex]

[tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=1-{P(z<2.56)-P(z<-2.56)}[/tex]

Refer the z table

[tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=1-{0.9948-0.0052}[/tex]

[tex]P(\bar{x}<62-2.2) or P(\bar{x}>62+2.2)=0.0104[/tex]

Hence The probability that the sample mean would differ from the population mean by greater than 2.2 kilograms is 0.0104 .

0.0104 or 1.04% is the probability that the sample mean weight of 195 adults will differ from the population mean .

We have to  use the central limit theorem.

Given the mean weight (62 kg) and variance (144 kg²) of the population, we first calculate the standard deviation:

Step 1: Calculate the population standard deviation.

Step 2: Calculate the standard error of the mean.

Step 3: Find the z-scores corresponding to ±2.2 kg difference from the mean.

Step 4: Use the standard normal distribution to find the probabilities.

Step 5: Calculate the total probability.

Therefore, the probability that the sample mean differs from the population mean by more than 2.2 kg is approximately 0.0104 or 1.04%.

Answer:

x = 2.4 or 12/5 or 2 and 2/5

Step-by-step explanation:

4(8x + 7) = 17x - 8

4 * 8x = 32x

4 * 7 = 28

32x + 28 = 17x - 8

-28             - 28

32x = 17x - 36

-17x       -17x

15x = -36

----      ----

15       15

x = 2.4 or 12/5 or 2 and 2/5

Hey!

---------------------------------------------------

Solution:

4(8x + 7) = 17x - 8

32x - 28 =  17x - 8

32x - 28 + 28 = 17x - 8 + 28

32x = 17x + (-36)

32x - 17x = 17x - (36) - 17x

15x = -36

15x/15 = -36/15

x = -36/15 or -12/5

---------------------------------------------------

Answer:

x = -12/5

---------------------------------------------------

Hope This Helped! Good Luck!

Answer: 100 miles, 4 gallons used

Step-by-step explanation:

Dist = rate * time = 50 mi/hr * 2 hr = 100 mi

Miles = gallons * miles/gallon. 100 = gallons*25mpg. Gallons = 100/25 = 4

Answer:

Mr. Chang will have 15714.90 dollars.

Step-by-step explanation:

p = 650

r = [tex]7.8/4/100=0.0195[/tex]

Number of periods or n = [tex]5\times4=20[/tex]

Future value formula is : [tex]p[\frac{(1+r)^{n}-1}{r} ][/tex]

Putting the values in formula we get;

[tex]650[\frac{(1+0.0195)^{20}-1}{0.0195} ][/tex]

= $15714.90

Hence, Mr. Chang will have 15714.90 dollars.

Answer with explanation:

Let us assume that the 2 functions are:

1) f(x)

2) g(x)

Now by definition of concave function we have the first derivative of the function should be strictly decreasing thus for the above 2 function we conclude that

[tex]\frac{d}{dx}\cdot f(x)<0\\\\\frac{d}{dx}\cdot g(x)<0[/tex]

Now the sum of the 2 functions is shown below

[tex]y=f(x)+g(x)[/tex]

Diffrentiating both sides with respect to 'x' we get

[tex]\frac{dy}{dx}=\frac{d}{dx}\cdot f(x)+\frac{d}{dx}\cdot g(x)\\\\[/tex]

Since each term in the right of the above equation is negative thus we conclude that their sum is also negative thus

[tex]\frac{dy}{dx}<0[/tex]

Thus the sum of the 2 functions is also a concave function.

Part 2)

The product of the 2 functions is shown below

[tex]h=f(x)\cdot g(x)[/tex]

Diffrentiating both sides with respect to 'x' we get

[tex]h'=\frac{d}{dx}\cdot (f(x)\cdot g(x))\\\\h'=g(x)f'(x)+f(x)g'(x)[/tex]

Now we can see the sign of the terms on the right hand side depend on the signs of the function's themselves hence we remain inconclusive about the sign of the product as a whole. Thus the product can be concave or convex.

Use the divergence theorem.

[tex]\vec F(x,y,z)=x\,\vec\imath+y\,\vec\jmath+5\,\vec k\implies\mathrm{div}\vec F(x,y,z)=2[/tex]

By the divergence theorem,

[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F\,\mathrm dV[/tex]

where [tex]R[/tex] is the region with boundary [tex]S[/tex].

Compute the latter integral in cylindrical coordinates, taking

[tex]\begin{cases}x=r\cos\theta\\y=y\\z=r\sin\theta\end{cases}\implies\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dy[/tex]

[tex]\displaystyle\iint_R2\,\mathrm dV=2\int_0^{2\pi}\int_0^1\int_0^{2-r\cos\theta}r\,\mathrm dy\,\mathrm dr\,\mathrm d\theta=\boxed{4\pi}[/tex]

Answer:

The equation does not have a real root in the interval [tex]\rm [0,1][/tex]

Step-by-step explanation:

We can make use of the intermediate value theorem.

The theorem states that if [tex]f[/tex] is a continuous function whose domain is the interval [a, b], then it takes on any value between f(a) and f(b) at some point within the interval. There are two corollaries:

Of course, in our case, we will make use of the first one.

First, we need to proof that our function is continues in [tex]\rm [0,1][/tex], which it is since every polynomial is a continuous function on the entire line of real numbers. Then, we can apply the first corollary to the interval [tex]\rm [0,1][/tex], which means to evaluate the equation in 0 and 1:

[tex]f(x)=x^3-3x+8\\f(0)=8\\f(1)=6[/tex]

Since both values have the same sign, positive in this case, we can say that by virtue of the first corollary of the intermediate value theorem the equation does not have a real root in the interval [tex]\rm [0,1][/tex]. I attached a plot of the equation in the interval [tex]\rm [-2,2][/tex] where you can clearly observe how the graph does not cross the x-axis in the interval.  

Answer:

About 139,764,705.9 users were added, on average, per year, from 1995 to 2012.

Step-by-step explanation:

The problem states that in june 2012, about 2.4 billion people used the internet - 100 times more than the number of people who used it in June 1995.

So, in June 2012, 2.4 billion people used the internet.

In June 1995, [tex]\frac{2,400,000,000}{100} = 24,000,000[/tex] = 24 million people used the internet.

About how many people were added on average each year from 1995 to 2012?

[tex]2012 - 1995 = 17[/tex]. There were 17 years between 1995 and 2012.

[tex]2,400,000,000 - 24,000,000 = 2,376,000,000[/tex]. There were  2,376,000,000 internet users added during this 17-year period. To find this number per year, we solve the following rule of three:

1 year - x users

17 years - 2,376,000,000 users

[tex]17x = 2,376,000,000[/tex]

[tex]x = \frac{2,376,000,000}{17}[/tex]

[tex]x = 139,764,705.9[/tex]

About 139,764,705.9 users were added, on average, per year, from 1995 to 2012.

Answer:

a) There is a 83.06% probability that all orders are shipped on time.

b) There is a 15.90% probability that exactly one order is not shipped ontime.

c) The probability of at least two orders being late is 1.02% + 0.02% = 1.04%.

Step-by-step explanation:

Probability:

What you want to happen is the desired outcome.

Everything that can happen iis the total outcomes.

The probability is the division of the number of possible outcomes by the number of total outcomes.

In our problem, there is:

-A 6% probability that a customer's order is not shipped on time.

-A 94% probability that a customer's order is shipped on time.

We have these following orders:

O1 - O2 - O3.

(a) What is the probability that all are shipped on time?

The probabilities that each order is shipped on time are O1 = 0.94, O2 = 0.94 and O3 = 0.94. So:

[tex]P = (0.94)^{3}[/tex] = 0.8306

There is a 83.06% probability that all orders are shipped on time.

(b) What is the probability that exactly one is not shipped ontime?

The order's can be permutated. What this means? It means that we can have O1 late and O2,03 on time, O2 late and O1,O3 on time and O3 late and O1, O2 on time. We have a permutation of 3 elements(the orders) with 2 and 1 repetitions(2 on time and one late).

The probability that an order is late is:

[tex]P = (0.94)^{2}(0.06)[/tex] = 0.053 for each permutation

Considering the permutations:

[tex]P = 0.053*p^{3}_{2,1} = 0.053\frac{3!}{2!*1!} = 0.053*3 = 0.1590[/tex]

There is a 15.90% probability that exactly one order is not shipped ontime.

(c) What is the probability that two or more orders are not shipped on time?

P = P1 + P2, where P1 is the probability that two orders are late and P3 is the probability that all three orders are late.

P1

Considering the permutations, the probability that two orders are late is:

[tex]P_{1} = p^{3}_{2,1}*(0.94)*(0.06)^{2} = 3*(0.94)*(0.06)^{2} = 0.0102[/tex]

There is a 1.02% probability that two orders are late

P2

[tex]P_{2} = (0.06)^3 = 0.0002[/tex]

There is a 0.02% probability that all three orders are late.

The probability of at least two orders being late is 1.02% + 0.02% = 1.04%.

Answer:

5mL would be required to provide a patient with 0.25 mg of the drug substance.

Step-by-step explanation:

The problem states that an injectable solution contains 25μg of a dug substance in each 0.5 mL, and asks how many milliliters would be required to provide a patient with 0.25 mg of the drug substance.

So, the first step is the conversion of 25ug to mg, since the problem asks the answer in mg.

Each mg has 1000ug. So

1mg - 1000ug

xmg - 25ug

1000x = 25

[tex]x = \frac{25}{1000}[/tex]

x = 0.025 mg

It means that each 0.5mL of the solution contains 0.025mg of the drug. How many milliliters would be required to provide a patient with 0.25 mg of the drug substance.

0.5mL - 0.025mg

xmL - 0.25mg

0.025x = 0.5*0.25

[tex]x = \frac{0.5*0.25}{0.025}[/tex]

x = 10*0.5

x = 5mL

5mL would be required to provide a patient with 0.25 mg of the drug substance.

Final answer:

To provide a patient with 0.25 mg of a drug substance, 5 mL of the injectable solution is required when the solution concentration is 25μg per 0.5 mL.

Explanation:

If an injectable solution contains 25μg of a drug substance in each 0.5 mL, the question asks how many milliliters would be required to provide a patient with 0.25 mg of the drug substance. First, it is important to convert 0.25 mg to micrograms (μg) because the concentration of the drug is given in micrograms. Knowing that 1 mg = 1000 μg, we have:

0.25 mg = 0.25 × 1000 μg = 250 μg.

Next, if 25 μg is in 0.5 mL, we find how many times 25 μg goes into 250 μg to determine the volume needed:

250 μg / 25 μg/mL = 10 times

Since 25 μg is contained in 0.5 mL:

10 × 0.5 mL = 5 mL.

Therefore, 5 mL of the injectable solution is required to provide a patient with 0.25 mg of the drug substance.

Answer:

The equation that represents the line passing through the point (2, -4) with a slope of one half is

[tex]f(x) = \frac{1}{2}x - 5[/tex]

Step-by-step explanation:

The equation of a line can be described by a first order equation in the following format:

[tex]f(x) = ax + b[/tex]

In which a is the slope of the line.

Solution:

The line slope is [tex]\frac{1}{2}[/tex], so [tex]a = \frac{1}{2}[/tex].

The equation of the line now is:

[tex]f(x) = \frac{1}{2}x + b[/tex]

The problem states that the line passes through the point(2,-4). This means that when x = 2, f(x) = -4

So:

[tex]f(x) = \frac{1}{2}x + b[/tex]

[tex]-4 = \frac{1}{2}*(2) + b[/tex]

[tex]-4 = 1 + b[/tex]

[tex]b = -5[/tex]

So, the equation that represents the line passing through the point (2, -4) with a slope of one half is

[tex]f(x) = \frac{1}{2}x - 5[/tex]