The rate of growth in the exponential function is 1.98%
Exponential FunctionAn exponential function is defined by the formula f(x) = aˣ, where the input variable x occurs as an exponent. The exponential curve depends on the exponential function and it depends on the value of the x.
To solve this problem, we have to model the growth in an exponential function.
Population in 1997 = 2.12 million
Population in 2006 = 2.53 million
Time = 9
Since this is an exponential growth function, we can write this as;
2.53 = 2.12(1 + r)⁹
where r = rate of growth
2.53 = 2.12(1 + r)⁹
r = 0.0198
r = 1.98%
The rate of growth is 1.98%
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A service club agrees to donate at least 500 hours of community service. A full member is to give 4 hours, and a pledge gives 6 hours.
(a) Write the inequality that expresses this information.
(b) Graph the inequality.
(a) 4x + 6y ≥ 500 , (b) The graph of this inequality is a line with a slope of 4/6 and a y-intercept of 500/6.
(a) The inequality expresses the total hours of community service that must be given, which is at least 500 hours. The 4x term represents the hours given by the full members, and the 6y term represents the hours given by the pledges. The ≥ symbol means that the sum of these two terms must be greater than or equal to 500.
(b) To graph the inequality, you first need to find the y-intercept, which is 500/6. Then you need to find the slope, which is 4/6. This means that the line has a positive slope and passes through the point (0, 500/6). The graph of the inequality is a line that passes through this point and has a positive slope of 4/6.
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Use algebraic rules of equations to predict the solution type to the system of equations
The solution type to the system is one solution or consistent system
How to predict the solution type to the system?From the question, we have the following parameters that can be used in our computation:
f(x) = [x + y = 4
[y = 2x - 1
Substitute y = 2x - 1 in the equation x + y= 4
So, we have the following representation
x + 2x - 1 = 4
Evaluate the like terms
3x = 5
So, we have
x = 5/3
Substitute x = 5/3 in y = 2x - 1
So, we have the following representation
y = 2 * 5/3 - 1
Evaluate
y = 2.33
Because x and y have real values
Then, the solution type is a consistent system
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2. JOY leather, a manufacturer of leather Products, makes three types of belts A, B and C which are processed on three machines M1, M2 and M3. Belt A requires 2 hours on machine (M1) and 3 hours on machine (M2) and 2 hours on machine (M3). Belt B requires 3 hours on machine (M1), 2 hours on machine (M2) and 2 hour on machine (M3) and Belt C requires 5 hours on machine (M2) and 4 hours on machine (M3). There are 8 hours of time per day available on machine M1, 10 hours of time per day available on machine M2 and 15 hours of time per day available on machine M3. The profit gained from belt A is birr 3.00 per unit, from Belt B is birr 5.00 per unit, from belt C is birr 4.00 per unit. What should be the daily production of each type of belt so that the profit is maximum?
a) Formulate the problem as LPM
b) Solve the LPM using simplex algorithm.
c) Interpret the shadow prices
The number of belts produced must be positive: X1, X2, X3 >= 0
The shadow price of a constraint can be understand as the change in the objective function value for a unit change in the right-hand side (RHS) value of the constraint, with all other variables held constant.
What are constraints?A constraint is a condition of an optimization problem that should be satisfied the condition.
Here we can formulate this problem as a linear programming model (LPM):
Consider X1 be the number of belt A produced per day
Let X2 be the number of belt B produced per day
Let X3 be the number of belt C produced per day
The objective function is to maximize the profit, that is;
3X1 + 5X2 + 4X3.
The constraints are as follows:
The time spent on machine M1 must be less than or equal to 8 hours per day then;
2X1 + 3X2 <= 8
The time spent on machine M2 must be less than or equal to 10 hours per day then;
3X1 + 2X2 + 5X3 <= 10
The time spent on machine M3 must be less than or equal to 15 hours per day then we have
2X1 + 2X2 + 4X3 <= 15
Thus number of belts produced must be positive:
X1, X2, X3 >= 0
To solve this LPM using the simplex algorithm, you can follow these steps:
w In standard form, the objective function is to minimize the cost, and all constraints are in the form of "less than or equal to" a certain value.
The pivot row is the row that corresponds to the smallest positive ratio between the right-hand side (RHS) value and the corresponding pivot column element the constraint 2X1 + 3X2 ≤ 8.
Now Interpret the shadow prices.
The shadow price of a constraint can be understand as the change in the objective function value for a unit change in the right-hand side (RHS) value of the constraint, with all other variables held constant.
It can help to understand the value of the constraints in the LPM and how they contribute to the optimal solution.
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G(x) = 4x - 5, find g(3)
Answer:
7
Step-by-step explanation:
g(3)=4(3)-5
g(3)=12-5
g(3)=7
Answer:
g(3) = 7
Step-by-step explanation:
Given function:
[tex]g(x)=4x-5[/tex]
To find g(x3) substitute x = 3 into the function:
[tex]\begin{aligned}x=3 \implies g(3)&=4(3)-5\\&=12-5\\&=7\end{aligned}[/tex]
Therefore:
g(3) = 7A store manager purchased a total of 15 pens and markers. Each pen cost $3.50, and each marker cost $1.75. If the manager spent a total of $38.50, how many pens did the store manager purchase?
The store manager purchases 7 pens which can be determined by the concept of equations.
What are equations?According to question store manager purchased a total of 15 pens and markers. So, let us assume the total number of pens =x and the total number of markers = y
The above statement gives us the equation as follows:
x + y=15
The statement each pen cost $3.50, and each marker cost $1.75 and the manager spent a total of $38.50 gives us the equation as follows:
3.50x +1.75 y=38.50
By solving both the equation on multiplying first eq by 3.50 gives
3.50x+3.50y=52.5
Subtracting the above two equations gives
1.75y=14
y=8
Putting value of y in first equation gives x=7
As we had assumed that the total number of pens = x
Hence, the total number of pens =7
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My question is what is the answer to this question when
7p+5 when p=12
Answer:89
Step-by-step explanation:math
Answer:89
Step-by-step explanation:7p+5
This means 7 *p+5
Where we see the p we put in 12
We get 7*12+5=89
Does the graph represent a proportional relationship?
graph with a line from 0 comma 0 and going through 10 comma 2
No, it is not proportional because the line does not go through (0, 0).
No, it is not proportional because the graph is curved.
Yes, it is proportional because it is a line that goes through (0, 0).
Yes, it is proportional because the x-axis and y-axis are numbered correctly.
pls help
Answer:
Yes, it is proportional because it is a line that goes through (0, 0).
Step-by-step explanation:
Answer:
Yes, it is proportional because it is a line that goes through (0, 0).
Step-by-step explanation:
Select all of the following equations that are true when 3.27 is used. Use number sense to help.
□ ÷ 1,000 = 0.0327
□ ÷ 102 = 0.327
□ ÷ 1 = 3.27
□ ÷ 10 = 0.327
□ ÷ 100 = 3.27
Answer:
The following equations are true when 3.27 is used:
÷ 1,000 = 0.0327
÷ 10 = 0.327
÷ 100 = 3.27
Note that the equations ÷ 102 and ÷ 1 are not true when 3.27 is used. The equation ÷ 102 is not true because 102 is not a multiple of 1,000, so dividing 3.27 by 102 will not result in a value that is close to 0. The equation ÷ 1 is not true because it is simply stating that 3.27 divided by 1 is equal to 3.27, which is true but not very interesting or useful.
Of the 24 fast-food businesses on Valley Mills Drive, the number that have a drive-up window, outside seating, or a pay telephone is summarized as follows:
18 have a drive-up window.
17 have outside seating.
8 have pay telephones.
12 have a drive-up window and outside seating.
6 have outside seating and a pay telephone.
5 have a drive-up window and a pay telephone.
4 have all three.
Find the number of businesses that have the following.
(a) a drive-up window and no outside seating
(b) a drive-up window and outside seating, but no pay telephone
(c) no pay telephone
(d) only a drive-up window
(e) outside seating or a pay telephone
By using a Venn diagram, we can conclude that:
a. 10 businesses have a drive-up window and no outside seating
b. 4 Businesses have a drive-up window and outside seating, but no pay telephone
c. 20 businesses have no pay telephone
d. 9 businesses have only a drive-up window
e. 8 businesses have outside seating or a pay telephone
A Venn diagram is a diagram consisting of overlapping circles to illustrate the logical relationship between two or more sets of items.
In the question, there are 3 sets of criteria that describe the condition of the fast-food businesses in Valley Mills Drive: whether or not there is a drive-up window, outside seating, or a pay telephone.
Let d denotes the number of businesses that have the three criteria.
If a denotes the number of businesses that have drive-up window and outside seating only, then:
drive-up window and outside seating = a + d
12 = a + 4
a = 8
If b denotes the number of businesses that have outside seating and a pay telephone only, then:
outside seating and a pay telephone = b + d
6 = b + 4
b = 2
If c denotes the number of businesses that have drive-up window and a pay telephone only, then:
drive-up window and a pay telephone = c + d
5 = c + 4
c = 1
If x denotes the number of businesses that have drive-up window only, then:
have a drive-up window = a + c + d + x
18 = 4 + 1 + 4 + x
18 = 9 + x
x = 9
If y denotes the number of businesses that have outside seating only, then:
have outside seating = a + b + d + y
17 = 4 + 2 + 4 + y
17 = 10 + y
y = 7
If z denotes the number of businesses that have pay telephone only, then:
Have pay telephone = b + c + d + z
8 = 2 + 1 + 4 + z
8 = 7 + z
z = 1
Then we can draw a Venn diagram to illustrate those numbers to get a better illustration in answering the questions.
a. a drive-up window and no outside seating = c + x = 1 + 9 = 10
b. a drive-up window and outside seating, but no pay telephone = a = 4
c. no pay telephone = x + y + a = 9 + 7 + 4 = 20
d. only a drive-up window = x = 9
e. outside seating or a pay telephone = y + z = 7 + 1 = 8
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The initial state of a queuing system (for example, when a restaurant first opens), is referred to as the ____ state. primary steady start-up transientintroductory
The initial state of a queuing system (for example, when a restaurant first opens), is referred to as the primary state.
In the queue system, what are the three states?1) Orderly queue, also known as FIFO (First In, First Out) or FCFS (First Come, First Serve).
2) Stack, also known as LCFS (Last Come First Serve), or LIFO (Last In First Out).
3) SIRO (Serve in Irregular Request).
What is the queuing system's steady state?Let P n(t) represent the probability of having n customers in the system at time t. This is the steady state of a queueing system, where the probability of the number of customers in the system is independent of t.
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suppose the relative risk of an outcome given exposure versus non-exposure is unity. determine the percent increased risk rounded to the nearest hundredth of a percent.
The percentage of higher risk is unaffected by exposure as relative risk in unity.
In order to determine the strength of the relationship between exposures (treatments or risk factors) and outcomes, relative risk is employed in the statistical analysis of data from ecological, cohort, medical, and intervention studies. For instance, in a study looking at how the drug apixaban affected the occurrence of thromboembolism, 8.8% of patients receiving a placebo developed the condition, whereas only 1.7% of patients receiving the medication did. This means that the relative risk is.19 (1.7/8.8), meaning that patients receiving apixaban had 19% the disease risk of patients receiving a placebo. [4] In this instance, apixaban lowers the probability of disease, making it a protective rather than a risk factor.
Values of relative risk can be understood as long as the relationship between the exposure and the result is assumed to be causal.
RR = 1 denotes that the outcome is unaffected by exposure.
When the risk of the outcome is less than 1, this is referred to as a "protective factor."
RR > 1 indicates that the exposure, a "risk factor," has an effect on the outcome's risk.
Therefore, the percentage of higher risk is unaffected by exposure as relative risk in unity.
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A. (f•g)(x)
B. (f•g)(-3)
f(x) = x^2+3x-2
g(x)= x-2
Answer: A. x^3 +x^2-8x+4 B. 10
Step-by-step explanation: (f•g)(x) is the same as f(x) • g(x)
A. (x^2+3x-2) • (x-2) = (x^3+3x^2-2x) -(2x^2+6x-4) -> x^3 +x^2-8x+4
B. plug in -3 for x -> (-3)^3 + (-3)^2 -8(-3) + 4 -> 10
Sunil drove 63.9823 km to visit his grandmother. When he reached his grandmother's house he noticed that the car had used up 3.31 litre of petrol. How many km can the car go in 1 litre?
The km per liter based on the information that Sunil can drive is 19.33 km.
How to illustrate the expression?An expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both.
In this case, Sunil drove 63.9823 km to visit his grandmother. When he reached his grandmother's house he noticed that the car had used up 3.31 litre of petrol.
Therefore, the km per liter will be:
= 63.9823 / 3.31
= 19.33 km per liters
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Determine the surface area of the cylinder. (Use π = 3.14)
net of a cylinder where radius of base is labeled 5 inches and a rectangle with a height labeled 4 inches
157 in2
219.8 in2
282.6 in2
314 in2
Answer:
282.6in^2
Step-by-step explanation:
Surface area of a cylinder = 2πrh+2πr^2
plugging in r=5 and h=4 gives
2π20+2π25
40π+50π = 90π
90*3.14=282.6in^2
A circular field has a diameter of 60 m. So Andi ran around the area 3 times. Then the distance traveled by Andi is .... meters.
Answer:
565.2 m------------------------------
One full circle is:
C = πdC = 3.14*60 m = 188.4 mThree times the same distance:
188.4 m * 3 = 565.2 mAnswer:
Distance travelled = 565.2 m
Step-by-step explanation:
Formula of circumference,
→ C = 2πr = π × d
The circumference of circle is,
→ π × d
→ (22/7) × 60
→ 1320/7
→ 188.4
Distance traveled after 3 rounds,
→ C × 3
→ 188.4 × 3
→ 565.2 m
Hence, Andi covered 565.2 m.
Which expression is equivalent to 8(6s+6)?
Answer:
48s + 48
Step-by-step explanation:
Find the unit rate.
1. A sloth moves 10 feet in 2 hours. It moves
2. The car travels 165 miles in 3 hours. It travels
feet per hour.
3. There are 48 bottles of water in 4 cases. There are
4. Juice cost $3.60 for 12 ounces. It costs
miles per hour.
per ounce.
bottles per case.
Step-by-step explanation:
1.
10 ft in 2 hours = 10/2 / 1 hour = 5 ft/h
2.
165 miles in 3 hours = 165/3 / 1 hour = 55 m/h or mph
3.
48 bottles in 4 cases = 48/4 / 1 case = 12 bottles per case
4.
$3.60 for 12 oz = 3.6/12 / 1 oz = $0.30 per ounce
May someone help me with this.
Step-by-step explanation:
since I don't see the options, I can only tell you what it is supposed to be and you need to pick the best fitting option :
the score of the test is simply
the number of correctly answered questions times 3 (as every correctly answered question is with 3 points).
the equation is therefore
y = R(x) = 3x
the independent variable x represents the number of correctly answered questions,
and the dependent variable is y, the test score, because the test score depends on the number of correctly answered questions.
so,
R(25) = 3×25 = 75, meaning 25 correctly answered questions × 3 results in the score of 75.
At Jalisco Electronics, assemblers are paid according to the following differential piece 4. rate scale: 1–20 units in a day, $4.50 each; 21–30 units, $5.50 each; and $7 each for every
unit over 30. Adrian Ortega assembled 70 units in one week. Find his gross pay.
The gross pay of Adrian Ortega for assembling 70 units of electronics in a week is $425
What is his gross pay?Gross pay refers to the total payment earned over a period of time.
1–20 units in a day = $4.50 each21–30 units = $5.50 eachOver 30 units = $7 eachAdrian Ortega assembled 70 units in one week
Pay for 1–20 units = $4.50 × 20
= $90
Pay for 21 - 30 units = $5.50 × 10
= $55
Pay for 30 - 70 = $7 × 40
= $280
Gross pay = $90 + $55+ $280
= $425
Therefore, Adrian Ortega is paid $425 in a week for assembling 70 electronics.
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you have 84 feet of fencing to enclose a rectangular plot that borders on a river. if you do not fence the side along the river, fidn the length and width of the plot that will maximise the area
Therefore the the length =21 and width =42 of the plot that will maximise the area is 882 [tex]ft^{2}[/tex] .
What is area?A region's size on a planar or curved surface can be expressed mathematically as its area. The term "surface area" refers to the area of a surface or the border of a three-dimensional object, whereas the term "plane area" refers to the area of a shape or planar lamina.
Here,
Here, take note that the shape is still a rectangle and the fencing is still 84 feet long.
As a result, the area is equal to xy and the sum of the sides NOT along the river is 84.
Thus, 2x + y = 84 and A = xy are the two equations.
We must find A as a function of x or y to determine the area with the largest value. My recommendation is to solve the first equation for y and substitute that value into the second equation.
A(x) = x and y = 84 - 2x (84-2x)
Currently, we must maximize A(x) = 84x - 2.
Recall that if we determine the vertex's x value by substituting, x = -b/(2a), we may find the vertex's y value.
Therefore, x = -84/(-4) = 21 ft because a = -2 and b = 84. If x = 21, y = 84 - 2(21) = 42
Thus, the dimensions are 21 by 42, and the maximum size is 21 by 42, or 882 square feet.
Therefore the the length =21 and width =42 of the plot that will maximise the area is 882 [tex]ft^{2}[/tex] .
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Which equation, when solved, results in a different value of x than the other three?
Equation of option (d) is having different value of x than other 3 equations.
So, option (d) is correct option.
How to solve equation?Write down the problem.Remember the order of operations: DMAS, which stands for Multiplication/Division, and Addition/SubtractionDo the multiplication.Do the addition and subtraction.Keep the variable at L.H.S and alone.To do this, just divide both sides of the equation by coefficient of x to find x.Solving equations
(a) (-7/8)x-(3/4)=20
adding 3/4 on both sides, we get
(-7/8)x = 20+(3/4)
multiplying (-8/7) on both sides, we get
x = -23.71
(b) (7/8)x+(3/4)=-20
subtracting 3/4 on both sides, we get
(7/8)x = -20-(3/4)
multiplying (8/7) on both sides, we get
x = -23.71
(c) 7(-1/8)x-(3/4)=20
adding 3/4 on both sides, we get
(-7/8)x = +20+(3/4)
multiplying (-8/7) on both sides, we get
x = -23.71
(d) (-7/8)×(-8/7)x-(3/4)=20×(-8/7)
⇒ x-(3/4) = -22.85
adding 3/4 on both sides, we get
x = -22.85+(3/4)
x = -22.10
So equation in option (d) has different value of x than all other equation.
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The profit, p(x), that Rivka’s Delivery Service makes from delivering packages between local businesses is P(x), where x is the number of $0.10 price increases for each delivery. Use the graph below to answer the question.
2.5 price hikes are often required to optimize profit.
How do you calculate the cost per unit?
To get the unit price of an item, we divide the price of a specific number of units by the number of units. If 12 ounces of soup cost $2.40, for instance, divide $2.40 by 12 ounces to get the unit price of soup, which is $0.20 per ounce.
x = the number of price hikes
y = Earnings
Additionally from the query, we have
starts at (0, 50)
expands to (2.5, 81.25)
Reduce through (6.531, 0)
It follows that
the most is (2.5, 81.25)
Remember that
x = the number of price hikes
Thus, we have
x = 2.5
Hence, the number of price increases is 2.5
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Hello I have to find the perimeter.
Answer:
10/3 mile
Step-by-step explanation:
perimeter = 4 × side
perimeter = 4 × 5/6 mile
perimeter = 20/6 mile
perimeter = 10/3 mile
Enola wants to invest $8,600.00 in a savings account that pays 5.1% simple interest.
How many years will it take for this investment to triple in value?
Round your answer to the nearest tenth of a year.
years for this investment to triple in value.
It will take _____ years for this investment to triple in value.
Step-by-step explanation:
To find the number of years it will take for Enola's investment to triple in value, we can use the formula for simple interest:
I = P * r * t
where I is the total interest earned, P is the principal (the initial amount invested), r is the interest rate, and t is the number of years.
Since the investment will triple in value, the total interest earned will be equal to two times the initial investment, or 2 * $8,600.00 = $17,200.00. Substituting these values into the formula and solving for t, we get:
$17,200.00 = $8,600.00 * 5.1% * t
t = $17,200.00 / ($8,600.00 * 5.1%) = 3.28 years
Rounding this value to the nearest tenth of a year, we get t = 3.3 years. Therefore, it will take approximately 3.3 years for Enola's investment to triple in value.
given the function that indicates the time it takes to double an investment compounded continiuosly at 6.35% then solve for t
10.9 years long will it take for $1000 to double at a 6.35% annual interest rate that is compounded every month.
Given that,
We have to find how long will it take for $1000 to double at a 6.35% annual interest rate that is compounded every month.
We know that,
P = $1000
r = 6.35% = 0.635
n=12
A = $2000
A = P(1+r/n[tex])^{nt}[/tex]
2000 = 1000(1+0.635/12[tex])^{12t}[/tex]
2= (1+0.635/12[tex])^{12t}[/tex]
log2= log(1+0.635/12[tex])^{12t}[/tex]
log2= 12t×log(1+0.635/12)
t=log2/12×log(1+0.635/12)
t≈10.9 years.
Therefore, 10.9 years long will it take for $1000 to double at a 6.35% annual interest rate that is compounded every month.
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A line passes through the point (4, -8) and has a slope of 5/2
Write an equation in slope-intercept form for this line.
help me pls this is due in 30 minutes
y=5/2x+24
Step-by-step explanation:
use the formula y-y1=m(x-x1)
5/2 symbolizes m
put in 4 for y1 and put in -8 for x1
y-(4)=(5/2)(x-(-8)
y-4=5/2(x+8)
y-4=5/2x+20
y=5/2x+24
Answer:
y = 5/2x -18
Step-by-step explanation:
You want the slope-intercept equation for the line with slope 5/2 through point (4, -8).
Point-slope formIt can work well to start with the point-slope equation and rearrange it to the desired form.
y -k = m(x -h) . . . . . . . line with slope m through point (h, k)
y -(-8) = 5/2(x -4) . . . . . line with slope 5/2 through point (4, -8)
Slope-intercept formAdd -8 and eliminate parentheses
y = 5/2x -10 -8
y = 5/2x -18 . . . . . . . . simplify
The desired equation is y = 5/2x -18.
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Let Q represent a mass (in grams) of carbon (^14C), whose half-life is 5,715 years. The quantity of carbon-14 present after t years is given by the following
Q = 11(1/2)^(t/5715)
a. Determine the initial quantity (when t=0) g
b. Determine the quantity present after 2000 years. (Round your answer to 2 decimal places).
c. What is the graph of the function over the interval t = 0 to t = 10,000
Answer:
(a) 11 g
(b) 8.63 g
(c) Attached
Step-by-step explanation:
Given function:
[tex]Q = 11 \left(\dfrac{1}{2}\right)^{\dfrac{t}{5715}}[/tex]
where:
Q = quantity of carbon-14 presentt = time (in years)Part (a)Substitute t = 0 into the given function to determine the initial quantity:
[tex]\begin{aligned} t=0 \implies Q &= 11 \left(\dfrac{1}{2}\right)^{\dfrac{0}{5715}}\\\\&=11\left(\dfrac{1}{2}\right)^0\\\\&=11\left(1\right)\\\\&=11\; \rm g\end{aligned}[/tex]
Part (b)Substitute t = 2000 into the given function to determine the quantity present after 2000 years:
[tex]\begin{aligned} t=2000 \implies Q &= 11 \left(\dfrac{1}{2}\right)^{\dfrac{2000}{5715}}\\\\&=11\left(0.784697887...\right)\\\\&=8.63\; \rm g\;\; \sf (2 \; d.p.)\end{aligned}[/tex]
Part (c)[tex]\begin{aligned} t=10000 \implies Q &= 11 \left(\dfrac{1}{2}\right)^{\dfrac{10000}{5715}}\\\\&=11\left(0.297346855...\right)\\\\&=3.27\; \rm g\;\; \sf (2 \; d.p.)\end{aligned}[/tex]
The graph of the function over the interval t = 0 to t = 10000 is attached.
What is the value of the missing exponent in the equation 7.4 ÷ 10□ = 0.074?
either 1
or 2
or 3
or 0
Answer:
The result is 0.074 ÷ 1000 = 0.00074, which means that 10□ = 1000, so the value of the missing exponent is 3.
Step-by-step explanation:
write the quadratic equation y=x2-6x+7 in vertex form. y=
Look at the diagram
Complete the following sentences.
Angle e= Angle , as they are alternate angles.
Angle f= Angle , as they are alternate angles.
Angles a, b and c add up to °, as they are on a straight line.
Therefore, angles e, f and b also add up to
Answer:
Angle e = Angle a, as they are alternate angles.
Angle f = Angle c, as they are alternate angles.
Angles a, b, and c add up to 180°, as they are on a straight line.
Therefore, angles e, f and b also add up to 180°.