1. By de Moivre's theorem,
[tex]\left(4\left(\cos\left(\dfrac{7\pi}9\right) + i \sin\left(\dfrac{7\pi}9\right)\right)\right)^3 = 4^3 \left(\cos\left(\dfrac{21\pi}9\right) + i \sin\left(\dfrac{21\pi}9\right)\right) \\\\ ~~~~~~~~ = 64 \left(\cos\left(\dfrac{7\pi}3\right) + i \sin\left(\dfrac{7\pi}3\right)\right) \\\\ ~~~~~~~~ = 64 \left(\cos\left(\dfrac\pi3\right) + i \sin\left(\dfrac\pi3\right)\right) \\\\ ~~~~~~~~ = 64 \left(\dfrac12 + i\dfrac{\sqrt3}2\right) \\\\ ~~~~~~~~ = \boxed{32 + 32\sqrt3\,i}[/tex]
2. First write the given number in exponential/trigonometric form.
[tex]z = 5-5\sqrt3\,i[/tex]
has modulus
[tex]|z| = \sqrt{5^2 + \left(-5\sqrt3\right)^2} = \sqrt{100} = 10[/tex]
and since it lies in the second quadrant of the complex plane, its argument is
[tex]\arg(z) = \pi + \tan^{-1}\left(-\dfrac{5\sqrt3}5\right) = \pi + \tan^{-1}\left(-\sqrt3\right) = \pi - \dfrac\pi3 = \dfrac{2\pi}3[/tex]
So, we have
[tex]z = 5 - 5\sqrt3\,i = 10 e^{i2\pi/3} = 10 \left(\cos\left(\dfrac{2\pi}3\right) + i \sin\left(\dfrac{2\pi}3\right)\right)[/tex]
Now we apply de Moivre's theorem again, and make sure to account for the multivalued-ness of the exponential function. For [tex]k\in\{0,1,2,3,4\}[/tex], the fifth roots of [tex]z[/tex] are
[tex]z^{1/5} = 10^{1/5} e^{i(2\pi/3 + 2\pi k)/5}[/tex]
[tex]k=0 \implies z^{1/5} = 10^{1/5} e^{i2\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{2\pi}{15}\right) + i \sin\left(\dfrac{2\pi}{15}\right)\right)}[/tex]
[tex]k=1 \implies z^{1/5} = 10^{1/5} e^{i8\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{8\pi}{15}\right) + i \sin\left(\dfrac{8\pi}{15}\right)\right)}[/tex]
[tex]k=2 \implies z^{1/5} = 10^{1/5} e^{i14\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{14\pi}{15}\right) + i \sin\left(\dfrac{14\pi}{15}\right)\right)}[/tex]
[tex]k=3 \implies z^{1/5} = 10^{1/5} e^{i20\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{4\pi}3\right) + i \sin\left(\dfrac{4\pi}3\right)\right)}[/tex] [tex]k=4 \implies z^{1/5} = 10^{1/5} e^{i26\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{26\pi}{15}\right) + i \sin\left(\dfrac{26\pi}{15}\right)\right)}[/tex]
You are on the Arithmetic Reasoning section.
Q: If 12 men are needed to run 4 machines. How many men are needed to
run 20?
A. 24
B. 48
C. 60
D. 80
Answer:that will be c 60
Step-by-step explanation: Because 12/4 but than the machines turn to 20 so 4x5 equals 20 so you got to do it to both sides
12x5=60 so its 60 men to run 20 machines
(sin alpha +cos alpha )2
Step-by-step explanation:
sin alpha^2 + 2sin alpha* cos alpha + cos alpha^2
sin alpha^2 + cos alpha^2 +2sin alpha* cos alpha
1+ 2sin alpha* cos alpha
1+sin2 alpha
A pair of shoes is reduced by 30% down to a price of £56
What was the original price of the shoes?
Answer:
To know the final result we have to make 30% of 56 dollars and add the result to the 56 dollars to know the original price.
Operations:
30% de 56 = (30x56)/100= 16,8 dollars.
56+16,8= 72,8 dollars (final result)
A pair of shoes is reduced by 30% down to a price of £56 then the original price of the pair of shoes was £80.
To find the original price of a pair of shoes using a mathematical approach. We are given that the shoes were reduced by 30% and their final price is £56. We will use algebraic equations to determine the original price of the shoes.
Let's denote the original price of the shoes as "P". Since the shoes were reduced by 30%, we can express this reduction as a decimal, which is 0.30 (30% = 30/100 = 0.30).
The amount of reduction is then given by 0.30 multiplied by the original price, which is 0.30P.
The final price of the shoes after the reduction is £56. We can set up an equation to represent this:
P - 0.30P = 56
Now, let's simplify the equation:
0.70P = 56
Next, we need to isolate "P" on one side of the equation. To do this, we divide both sides by 0.70:
[tex]\[ \frac{{0.70P}}{{0.70}} = \frac{{56}}{{0.70}} \][/tex]
This simplifies to:[tex]\[ P = \frac{{56}}{{0.70}} \][/tex]
Now, let's calculate the value of "P":
[tex]\[ P = \frac{{56}}{{0.70}} = 80 \][/tex]
So, the original price of the pair of shoes was £80.
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An account begins the year with a
$4000.00 balance. If the account
earns 3% simple interest, what is the
balance at the end of one year?
A. $4,120.00
C. $4,300.00
B. $4,003.00
D. $4,012.00
[tex]s = c(1 + in) \\ s = 4000(1 + 0.03 \times 1) [/tex]
[tex]s = 4120.00[/tex]
Option: AConsider the function f(x)=x
Describe the transformation of g(x)=2f(x-4)+2
The parent function ____ will undergo ____ transformations. The first transformation is a _____ reflect by a factor of ______. Next, the graph will shift ______ units to the ______. Finally, the graph will shift _____ units ____.
Please help me!
The parent function f(x) = x will undergo 3 transformations. The first transformation is vertically stretched by a factor of 2. Next, the graph will shift 4 units to the right. Finally, the graph will shift 2 units up
How to determine the transformation?The function is given as:
f(x) = x
This represents the parent function.
Start by stretching the function vertically by a factor of 2
This gives
f'(x) = 2x
Next, we shift the graph 4 units to the right
This gives
f''(x) = 2(x - 4)
Finally, we shift the graph 2 units up
This gives
f'''(x) = 2(x - 4) + 2
The number of transformations above is 3
Hence, the complete statement is:
The parent function f(x) = x will undergo 3 transformations. The first transformation is vertically stretched by a factor of 2. Next, the graph will shift 4 units to the right. Finally, the graph will shift 2 units up
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PLEASE HELP I will give 50 points! PLEASE ANSWER CORRECTLY
In a school, 10% of the students have green eyes. Find the experimental probability that in a group of 4 students, at least one of them has green eyes. The problem has been simulated by generating random numbers. The digits 0-9 were used. Let the number "9" represent the 10% of students with green eyes. A sample of 20 random numbers is shown. 7918 7910 2546 1390 6075 2386 0793 7359 3048 1230 2816 6147 5978 5621 9732 9436 3806 5971 6173 1430 Experimental Probability = [?]% =
Using it's concept, the experimental probability that in a group of 4 students, at least one of them has green eyes is of 0.45 = 45%.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes. An experimental probability is calculated considering the results of previous trials.
In this problem, there were 20 trials. The digit 9, which represents a student with green eyes, appears at least once in 9 of the the trials, hence the probability is:
p = 9/20 = 0.45 = 45%.
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The following scatterplot shows the percentage of the vote a candidate received in the 2016 senatorial elections
according to the voter's income level based on an exit poll of voters conducted by a news agency. The income
levels 1-8 correspond to the following income classes:
1 = Under $15,000; 2 = $15-30,000; 3 = $30-50,000; 4 = $50-75,000; 5 = $75-100,000;
6 = $100-150,000; 7 = $150-200,000; 8 = $200,000 or more.
Use the election scatterplot to the find the critical values corresponding to a 0.01 significance level used to test
the null hypothesis of ρs = 0.
A) -0.881 and 0.881
B) -0.881
C) -0.738 and 0.738
D) 0.881
The critical values corresponding to a 0.01 significance level used to test the null hypothesis of ρs = 0 is (a) -0.881 and 0.881
How to determine the critical values corresponding to a 0.01 significance level?The scatter plot of the election is added as an attachment
From the scatter plot, we have the following highlights
Number of paired observations, n = 8Significance level = 0.01Start by calculating the degrees of freedom (df) using
df =n - 2
Substitute the known values in the above equation
df = 8 - 2
Evaluate the difference
df = 6
Using the critical value table;
At a degree of freedom of 6 and significance level of 0.01, the critical value is
z = 0.834
From the list of given options, 0.834 is between -0.881 and 0.881
Hence, the critical values corresponding to a 0.01 significance level used to test the null hypothesis of ρs = 0 is (a) -0.881 and 0.881
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Jeff decided to purchase a house, his mortage is a 30-year loan at 4.1 % compounded monthly. The final purchase price was $150,000.00 Jeff put down 2808.00 at closing. Find Jeffs monthly payment
Answer:
$711.23
Step-by-step explanation:
We assume the entire closing cost went to reducing the principal of the loan. Then the amount borrowed was $147,192.
Monthly paymentThe amortization formula tells you the monthly payment.
A = P(r/12)/(1 -(1 +r/12)^(-12t))
P is the principal, r is the annual rate, and t is the number of years.
The monthly payment is ...
A = $147,192(0.041/12)/(1 -(1 +0.041/12)^-360) ≈ $711.23
Jeff's monthly payment is $711.23.
The wall, shown above in Figure 105.1401, has an overall length that is 37’ and 8 1/4" . Three receptacle outlets are to be installed in this wall. L2 is twice as long as L1 . L1 should be _____' _____''.
Since L₂ is twice as long as L₁, the length of L₁ should be 12' and 6 3/4".
How to calculate the length (L₁)?First of all, we would convert the value of the overall length in feet to inches as follows:
1 feet = 12 inches
37 × 8 1/4 feet = X inches
Cross-multiplying, we have:
X = 1809/4 inches.
Since L₂ is twice as long as L₁, we have:
L₂ = 2L₁
Also, the overall length is given by:
T = 2L₁ + 2L₂
T = L₂ + 2L₂
T = 3L₂
1809/4 = 3L₂
L₂ = 1809/4 × 1/3
L₂ = 1809/12
L₂ = 150.75''
L₂ = 12' and 6 3/4".
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Which value of w makes 14 = 11 + w/8 x 6 a true statemen?
choose 1 answer:
w = 1
w = 4
w = 16
w = 24
answer
is true statement w= 4
PLS HELP plsssssssssssssss
Answer:
a. 33
Step-by-step explanation:
a straight line is 180° so
3x = 180 - 81
3x = 99
to find x divide by 3
x = 99/3
x = 33
Find the measure of Angle BAC.
Applying the inscribed angle theorem, the measure of angle BAC is: 110°.
What is the Inscribed Angle Theorem?According to the inscribed angle theorem, the measure of angle BAC is half the measure of the intercepted arc, x + 50°.
x - 30 + x + 50 = 360
2x + 20 = 360
2x = 360 - 20
2x = 340
x = 170
m∠BAC = 1/2(x + 50)
Plug in the value of x
m∠BAC = 1/2(170 + 50)
m∠BAC = 110°
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What would the first step be in solving the equation 4/7x = 21
❄ Hi there,
the first step in solving this inequality is
multiplying (both sides) by 7[tex]\sf{\dfrac{4}{7}x\times7=21\times7}[/tex]. Look what happens on the left side!
[tex]\sf{4x=147}[/tex]
Multiplying both sides of an equation, or inequality, by the fraction's denominator, is a nice little trick when you need to get rid of fractions to solve in terms of a variable.
Finish solving for x by dividing both sides by 4 –
[tex]\sf{x=\dfrac{147}{4}}[/tex]
❄
PLS ANSWER BOTH PARTS I WILL GIVE BRAINLIEST PLSSS HURRRYYYY
Answer: slope is 1/3
Step-by-step explanation: make a graph of y=1/3x+9
A 1L IV contains 60meq of kcal. The IV was discontinued after 400ml has infused. How much lvl did the patient receive
If a 1L IV contains 60meq of kcal and the IV was discontinued after 400ml has infused, then the amount of IV received by the patient is 24meq of kcal.
Calculation for the Amount of IV
It is given that,
1L IV contains 60meq of kcal
⇒ 1000 mL of IV contains 60meq of kcal
⇒ 1 mL of IV will contain 60 / 1000 meq of kcal
As per the question,
The amount of IV infused in the patient = 400 mL
⇒ The amount of IV received by the patient in meq of kcal = 400 × (60 / 1000)
= 4 × 6
= 24 meq of kcal
Hence, the patient receives 24meq of kcal amount of IV.
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Using the principle of
mathematical induction show that 10^(2n-1 ) + 1 is divisible by 11 for all z
When [tex]n=1[/tex],
[tex]10^{2\cdot1 - 1} + 1 = 10^1 + 1 = 11[/tex]
which is of course divisible by 11.
Assume this holds for [tex]n=k[/tex], that
[tex]11 \mid 10^{2k - 1} + 1[/tex]
In other words,
[tex]10^{2k - 1} + 1 = 11\ell[/tex]
for some integer [tex]\ell[/tex].
Use this to show the claim is true for [tex]n=k+1[/tex].
[tex]10^{2(k+1) - 1} + 1 = 10^{2k + 1} + 1 \\\\ ~~~~~~~~~~~~~~~~~~~~ = 10^{2k+1} + \left(10^{2k-1} + 10^{2k-1}\right) + 1 \\\\ ~~~~~~~~~~~~~~~~~~~~ = \left(10^{2k+1} - 10^{2k-1}\right) + \left(10^{2k-1} + 1\right) \\\\ ~~~~~~~~~~~~~~~~~~~~ = 10^{2k-1} \left(10^2 - 1\right) + 11\ell \\\\ ~~~~~~~~~~~~~~~~~~~~ = 99\times10^{2k-1} + 11\ell \\\\ ~~~~~~~~~~~~~~~~~~~~ = 11\left(9\times10^{2k-1} + \ell\right)[/tex]
which is indeed divisible by 11. QED
On the off-chance you meant [tex]10^{2^n-1}+1[/tex], notice that [tex]2n-1[/tex] is odd for any integer [tex]n[/tex]. Similarly [tex]2^n-1[/tex] is odd for all [tex]n[/tex], so the above proof actually proves this automatically.
Solve Log6( x) = -2
Write an equation for the function graphed below
f(x) = 12(x - 1)/[(x + 2)(x - 3)] is the equation of the function graphed as in the figure. This can be obtained using the formula of f(x) = P(X)/Q(X) by vertical asymptotes and factors of the polynomial.
Write an equation for the function:Let where P(X) and Q(X) are polynomial function.
The function has the following:
From the given graph, we can say that the vertical asymptotes x = - 2 and x = 3, this means that at values of x = -3 or x = 2, the denominator becomes zero and the function becomes undefined.This means that, -3 and 2 are roots of the denominator. Thus, the equation of the denominator is:
⇒ Q(X) = (x + 2)(x - 3)
From the given graph, we can say that the zero is 3, P(X) has the factor (x - 3)
Let a be the leading coefficient of P(X)
⇒ P(X) = a(x - 1)
Now for finding the value of a we can rewrite the equation as,
⇒ f(x) = P(X)/Q(X)
⇒ f(x) = a(x - 1)/[(x + 2)(x - 3)]
⇒ f(0) = a(0 - 1)/[(0 + 2)(0 - 3)]
Since from the graph we can say that the y-intercept is (0, - 2), this means that when x = 0, y = -2, f(0) = - 2
⇒ - 2 = -a/[(2)(- 3)]
⇒ a = - 2 × 2 × 3
⇒ a = - 12
By putting a = -12 in f(x) we get,
⇒ f(x) = 12(x - 1)/[(x + 2)(x - 3)]
Hence f(x) = 12(x - 1)/[(x + 2)(x - 3)] is the equation of the function graphed as in the figure.
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Uncle Bill has planted fruits in the farm and 7/12 is watermelon.2/3 of the watermelon section is ruined from the storm.The rest is planted with lemon which isn’t ruined.What is the fraction of his farm which isn’t ruined?
The fraction of his farm which isn’t ruined is 11/18
Fractions and proportionFractions are written as a ratio of two integers. The given question will be solved using the method of proportion.
If Uncle Bill has planted fruits in the farm and 7/12 is watermelon, and 2/3 of the watermelon section is ruined from the storm, the fraction ruined from the storm is;
Fraction ruined from the storm = 2/3* (7/12)
Simplify to its lowest term to have:
Fraction ruined from the storm = 7/18
Determine the fraction of the firm that is not ruined
Fraction not ruined = 1 - 7/18
Fraction not ruined = 11/18
Hence the fraction of his farm which isn’t ruined is 11/18 if uncle Bill has planted fruits in the farm and 7/12 is watermelon.
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Which table represents exponential growth? A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 6, 8. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 8, 16. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 7, 11. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 6, 10.'
The table that represents exponential growth is the second table.
What is exponential growth?Exponential growth simply means a process that increases quantity over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself.
Exponential growth is the pattern of data which shows sharper increases over time. In finance, compounding creates exponential returns and the savings accounts with a compounding interest rate can show exponential growth.
In this case, the second table described represents an exponential function. It has a common ratio of 4, meaning that each y-value is multiplied by 4 to get to the next value. Here, every time the Xs increase by 1, the Ys increase by a multiple of 4.
One of the best examples of exponential growth is observed in bacteria. Here, it takes bacteria roughly an hour to reproduce through prokaryotic fission. This illustrates exponential growth.
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Answer: B
Step-by-step explanation:
area of regular polygon?
[tex]\huge\purple{Hi!}[/tex]
The area of a regular polygon is one-half the product of its apothem and its perimeter.
Find the value of B - A if the graph of Ax + By = 3 passes through the point (-7,2), and is parallel to the graph of x + 3y = -5
Answer: -6
Step-by-step explanation:
[tex]x+3y=-5\\\\3y=-x-5\\\\y=-\frac{1}{3}x-\frac{5}{3}[/tex]
So, since parallel lines have the same slope, the slope of the line we need to find is -1/3.
Substituting into point-slope form, the equation is
[tex]y-2=-\frac{1}{3}(x+7)\\[/tex]
Converting to the required form,
[tex]y-2=-\frac{1}[3}x-\frac{7}{3}\\\\\frac{1}{3}x+y=-\frac{1}{3}\\\\-3x-9y=3[/tex]
So, B-A is equal to -6.
help me find the answer that bests best
represents this relationship
Answer:
c
Step-by-step explanation:
consider points (0,2) and (-4,0)
slope=(0-2)/(-4-0)=-2/-4=1/2
y-intercept=2
y=mx+c
y=1/2 x+2
Answer:
y=1/2x+2
Step-by-step explanation:
. Last weekend, Michael
drove to his friend's house.
When he left, he noticed
that the fuel gauge in his
car indicated that his gas
tank was 4 full. When he
returned home, the fuel
gauge in his car indicated
that his gas tank was ½ full.
If the gas tank holds 24
gallons, how many gallons
did Michael use on his
drive?
It can be said that the total amount of fuel that was used by Michael when he was away to his friends was 12 gallons.
How to solve for the quantity.The question said that the tank was full before he left to his friends. This was after he checked the gauge of the gas tank.
The question says that the required quantity needed to fill up this tank is 24 gallon. Hence if it is full, then the total gallon it had before he made this trip was 24 gallon.
Then we are told that he went ahead to check the gauge when he finished and was at home. The quantity of fuel he had at this time was 1/2 of what was there.
This would be 1/2 * 24
12
Hence he made use of 12 gallons when he was away.
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Find a formula for the nth term in this
arithmetic sequence:
a₁ = 7, a2 = 4, a3 = 1, a4 = -2, ...
an
=
[?]n +
Answer:
[tex]a_{n} = -3n + 10[/tex]
Step-by-step explanation:
The arithmentic function formula is:
[tex]a_{n} = a_{1} + d(n-1)[/tex]
Let's plug in what we know. [tex]a_{1}[/tex] (the first term in our sequence) is 7. d (the common difference) is -3. This means we are subtracting 3 every time.
Plugging that in to the formula, we get
[tex]a_{n} = 7 -3(n-1)[/tex]
Now we distribute and combine like terms.
[tex]a_{n} = 7 - 3n +3[/tex]
[tex]a_{n} = 10 - 3n[/tex]
Change the order to fit the format and you get
[tex]a_{n} = -3n + 10[/tex]
Trapezoid FGHJ is inscribed in P as shown. If FG = 24, JH = 10, and the diameter of P is 26, find the area of the trapezoid. (Hint: draw in auxiliary lines). Round to the nearest tenth if necessary.
Answer:
221
Step-by-step explanation:
If the diameter is 26, the radius is 13 (which is the same as the perpendicular from P to JH)
So, the area is
[tex]\frac{1}{2}(13)(24+10)=221[/tex]
You are volunteering to help with the soccer team's Valentine's Day fundraiser. Each 16-ounce bag of nuts the team sold must include at least 60% chocolate-covered nuts inside.
However, instead of receiving a shipment of separated plain nuts and chocolate nuts, they delivered two large containers of mixed nuts. The first says it is 50/50 plain and chocolate covered. The second contains 80% chocolate-covered nuts.
The team is dismayed, but you come up with a solution. You suggest combining
ounces* of the 50/50 nuts with
ounces* of the 80% chocolate-covered nuts, to create the 60% mixture required for each bag.
*estimate
Then Bob, another student on the team says, "Wait, we promised at least 60%. So if we just do half and half, won't we be giving them at least 60%?"
Bob
correct.
Proportionately, if they do half and half of 50% and 80% of the two bags, Bob is correct.
What is a proportion?A proportion is a mathematical measure that compares two variables or numbers.
Proportions can be stated in percentages or ratios, as decimals or fractions.
Data and Calculations:Content of 16-ounce bag of nuts = 60% chocolate-covered nuts
Content of first bag = 50/50 plain and chocolate-covered nuts
Content of second bag = 80% chocolate-covered nuts
A mixture of half of 50% bag with half of 80% bag will result in 65% (50% + 80%)/2.
Thus, since the promise for each 16-ounce bag is for at least 60% chocolate-covered nuts, Bob is correct.
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Question Completion:Is Bob correct?
Decreasing the number of years of a loan decreases the amount of interest repaid over the term of the loan. Suppose a dental hygienist has the option of a 30-year loan or a 25-year loan of $345,000 at an annual interest rate of 3.75%.
(a)
a) Calculate the monthly payment (in dollars) for each loan. (Round your answers to the nearest cent.)
30-year loan
25-year loan
b) Calculate the savings in interest (in dollars) by using the 25-year loan. (Round your answer to the nearest cent.)
Answer:
a) 30 yr: $1597.75; 25 yr: $1773.75
b) $43065.00
Step-by-step explanation:
The monthly payment for each loan can be found using the amortization formula, or a spreadsheet or calculator. The shorter 25-year loan has fewer and larger payments, but the net result is less interest paid.
a)The attached calculator shows the monthly payments to be ...
30 year loan: $1597.75 monthly
25 year loan: $1773.75 monthly
b)The number of payments is the product of 12 payments per year and the number of years. The total repaid is the monthly payment times the number of payments.
The difference in amounts repaid is the difference in interest charged.
360×1597.75 -300×1773.75 = $43065 . . . . savings using 25-year loan
__
Additional comment
The monthly payment on a loan of principal P at annual rate r for t years is ...
A = P(r/12)/(1 -(1 +r/12)^(-12t))
Some calculators and all spreadsheets have built-in functions for calculating this amount.
The total cost to rent 5 chairs and 3 tables is $27. The total cost to rent 2 chairs and 12 tables is $81. What is the cost to rent each chair and each table
The cost to rent each chair is $1.5 and cost to rent each table is $6.5
Applications of systems of linear equationsFrom the question, we are to determine the cost to rent each chair and each table
Let c represent chair
and
t represent table
From the given information,
The total cost to rent 5 chairs and 3 tables is $27
That is,
5c + 3t = 27 ------------ (1)
Also,
The total cost to rent 2 chairs and 12 tables is $81
That is,
2c + 12t = 81 ---------- (2)
Now, solve the equations simultaneously
5c + 3t = 27 ------------ (1)
2c + 12t = 81 ---------- (2)
Multiply equation (1) by 2 and multiply equation (2) by 5
2 × [5c + 3t = 27 ]
5 × [2c + 12t = 81 ]
10c + 6t = 54 ------------- (3)
10c + 60t = 405 ------------- (4)
Subtract equation (4) from equation (3)
10c + 6t = 54
10c + 60t = 405
---------------------------
-54t = -351
t = -351/-54
t = 6.5
Substitute the value of t into equation (2)
2c + 12t = 81
2c + 12(6.5) = 81
2c + 78 = 81
2c = 81 - 78
2c = 3
c = 3/2
c = 1.5
∴ The cost of chair is $1.5 and cost of table is $6.5
Hence, the cost to rent each chair is $1.5 and cost to rent each table is $6.5
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A tumor is injected with 0.9 grams of Iodine-125, which has a decay rate of 1.15% per day. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days.
With the use of formula, the amount of Iodine-125 that would remain in the tumor after 60 days is 0.45 grams
Word Problem Leading to Exponential FunctionFirst analyze the problem and represent them in exponential function. The decay rate is different from increase rate with minus and plus sign.
Given that a tumor is injected with 0.9 grams of Iodine-125, which has a decay rate of 1.15% per day. Let
I = initial amount injected = 0.9 gramsR = decay rate = 1.15%t = number of days = 60 daysA = the remaining amount of Iodine - 125An exponential model representing the amount of Iodine-125 remaining in the tumor after t days will be
A = I( 1 - R%)^t
Let us use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days by substituting all the given parameters into the formula
A = 0.9 ( 1 - 1.15/100)^60
A = 0.9 ( 1 - 0.0115)^60
A = 0.9 ( 0.9885)^60
A = 0.9 x 0.4995
A = 0.45 grams
Therefore, the amount of Iodine-125 that would remain in the tumor after 60 days is 0.45 grams
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