The domain restriction for f(x)/g(x) is x ≠ 0 or -3
How to find the domain restriction for f(x)/g(x)?The domain of a function is the set of all input values (or "independent variable") for which the function is defined and produces a valid output (or "dependent variable").
In other words, it is the set of all valid input values for which the function will produce a real output and not give an error or undefined result.
But the domain restriction is the set of values that are not allowed as input for the function.
f(x)/g(x) = (x+26)/(14x²+42x)
In order to find the Domain restriction for f(x)/g(x), equate the denominator g(x) to zero and solve for x. That is:
14x²+42x = 0
14x(x + 3) = 0
14x = 0 or x + 3 = 0
x = 0 or x = -3
Thus, the Domain restriction is x ≠ 0 or -3
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A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 47 and 57 minutes. Find the probability that a given class period runs between 50.5 and 51 minutes?
The probability would be length of the interval [50.75, 51.25] divided by the length of the interval [47.0, 57.0] which is 0.05
Probability :Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.
Now we have given :-the lengths of her classes are uniformly distributed between 47 and 57 minutes.
so probability hat a given class period runs between 50.5 and 51 minutes=
(51-50.5)/(57-47)
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The probability would be length of the interval [50.75, 51.25] divided by the length of the interval [47.0, 57.0] which is 0.05
Probability :Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.
Now we have given :-the lengths of her classes are uniformly distributed between 47 and 57 minutes.
so probability hat a given class period runs between 50.5 and 51 minutes= (51-50.5)/(57-47) = 0.05
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Write an equivalent expression by distributing the sign outside the parentheses:
- (-9c- 0.8d - 1)
Answer:
9c + 0.8d + 1
Step-by-step explanation:
- (- 9c - 0.8d - 1) ← multiply each term in the parenthesis by - 1
= 9c + 0.8d + 1
Enter deg after any value that is in degrees.
The concept of linear pairs from the given diagram shows that; GKH = 90°
How to identify linear pairs?A linear pair is formed when two straight lines intersect to form two angles that are adjacent to each other and are on a straight line.
Now, what this means is that ∠F KG and ∠GKH will sum up to 180 degrees because they are linear pairs from the question.
Due to the fact that ∠F KG = 90°, then we can say that;
∠F KG + ∠GKH = 180°
90° + ∠GKH = 180°
∠GKH = 180° - 90°
∠GKH = 90°
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Six years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ?
A. 11/P + 6
B. P/11 +6
C. 17 - P/6
D. 17/P
E. 11.5P
option A is correct-Ben is 11/P + 6 old now in terms of P .
Six years ago,
Anita was P times as old as Ben was.
Let Ben's age now be B.
let Anita's age now be A.
linear equation in two variables is as follows,
(A-6) = P(B-6)
we know A = 17,put the value of A,
17 - 6 = P(B - 6)
11 = P(B - 6)
11/P = B- 6
B = 11/P + 6
therefore,
Ben is 11/P + 6 old now in terms of P .
linear equation
A linear equation in two variables is of the form Ax + By + C = 0, in which A and B are the coefficients, C is a constant term, and x and y are the two variables, each with a degree of 1.
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x^5-5x^3+4x=0 solve the equation
Answer:
x
=
0
,
x
=
1
,
x
=
2
,
x
=
−
1
, and
x
=
−
2
Explanation:
Start off by factoring out an
x
as follows
x
(
x
4
−
5
x
2
+
4
)
=
0
We can factor
x
4
−
5
x
2
+
4
Doing so we have
x
(
x
2
−
4
)
(
x
2
−
1
)
=
0
So in order for this to be equal to zero
x
=
0
OR
x
2
−
4
=
0
OR
x
2
−
1
=
0
Well
x
=
0
x
2
=
4
x
=
±
√
4
=
±
2
x
2
=
1
x
=
±
√
1
=
±
1
consider the following balls-and-bin game. we start with one black ball and one white ball in a bin. we repeatedly do the following: choose one ball from the bin uniformly at random, and then put the ball back in the bin with another ball of the same color. we repeat until there are n balls in the bin. let x
The number of white balls is equally likely to be any number between 1 and n − 1.
This problem can be solved by considering the number of ways to get a certain number of white balls in the bin. Since each ball can either be white or black, there are 2^n possible sequences of n balls in the bin.
Now consider the number of ways to get exactly k white balls in the bin. It can be done in the following way:
Choose k white balls out of n balls, which can be done in C(n, k) ways, and then arrange the k white balls and n-k black balls in (n-k + k)!/(n-k)!k! ways.
Therefore, the number of ways to get exactly k white balls in the bin is C(n, k)(n-k + k)!/(n-k)!k! = C(n, k)(n!)/(n-k)!k!.
Since each sequence of balls is equally likely, it follows that the number of white balls is equally likely to be any number between 1 and n - 1.
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--The given question is incomplete; the complete question is
"Consider the following balls-and-bin game. We start with one black ball and one white ball in a bin. We repeatedly do the following: choose one ball from the bin uniformly at random, and then put the ball back in the bin with another ball of the same colour. We repeat until there are n balls in the bin. Show that the number of white balls is equally likely to be any number between 1 and n − 1."--
A plant is already 10.25 meters tall, and it will grow 5 centimeters every month. The plant's height, H (in meters), after x months is given by the following
function.
H(x)=0.05x+10.25
What is the plant's height after 30 months?
On solving the function H(x) = 0.05x + 10.25, the plant's height after a period of 30 months is obtained as 11.75 m.
What is a function?
In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
The height of the plant is = 10.25 m
The rate at which plant grows is = 5 cm/month or 0.05m/month
The function for plant's height is given as - H(x) = 0.05x + 10.25
To find the plant's height after 30 months, substitute the value of x with 30 in the function -
H(30) = 0.05(30) + 10.25
Use the arithmetic operation of multiplication -
H(30) = 1.5 + 10.25
Use the arithmetic operation of addition -
H(30) = 11.75
Therefore, the plant's height after 30 months will be 11.75 m.
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5. Nicolet is fencing off a rectangular pen for her chickens. She has 80 feet of fencing and will use all of it. The area of the pen, A(x), is a function of its length, x, in feet.
The domain of the area function is given by the following interval:
(0, 40).
How to obtain the domain of the function?The domain of a function is the set composed by all the input values that the function accepts.
On the graph, the domain of the function is given by the values of x.
Considering the graph, the domain is given by the following interval:
(0, 40).
It is an open interval because if x = 0, then the length is of zero, and if x = 40, then the width is of zero, meaning that these dimensions are not possible.
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Divide the following numbers in scientific notation and express the result in scientific notation.
Answer:
8.00
Step-by-step explanation:
First simplify the exponents 10^15 is 1000000000000000 or 1e+15 then 10^4 is 10000. Then multiply: 1e+15 x 3 = 3e+15 ------ 10000 x 7 = 70000. When you divide 3e+15 by 70000 you get 8.115 or 8 rounded to the decimal place.
Lisa is framing a rectangular painting. The length is three more than twice the width. She uses 30 inches of framing material. What is the length of the painting?
Answer:
11
Step-by-step explanation:
picture is too blurry, but here is the process to solve it
L = 3+2w
total length = 30 inches = 2L +2w
so 2w=30-2L
L= 3+2w = 3+30-2L
so 3L =33
L =11
NO LINKS!!!
60. If a bond earns 5% interest per year compounded continuously, how many years will it take for an initial investment of $100 to $1000?
Answer:
46.05 years
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]
Given:
A = $1000P = $100r = 5% = 0.05Substitute the given values into the continuous compounding formula and solve for t.
[tex]\implies 1000=100e^{0.05t}[/tex]
[tex]\implies 10=e^{0.05t}[/tex]
[tex]\implies \ln 10=\ln e^{0.05t}[/tex]
[tex]\implies \ln 10=0.05t\ln e[/tex]
[tex]\implies \ln 10=0.05t[/tex]
[tex]\implies t=\dfrac{1}{0.05}\ln 10[/tex]
[tex]\implies t=20\ln 10[/tex]
[tex]\implies t=46.0517018... \rm years[/tex]
Therefore, it will take 46.05 years for the initial investment of $100 to reach $1000.
HELPPPPP PLEASEEEE URGENT
The table 1 represent the function From the given figure.
What is function ?
The function in mathematical terms is the mapping of each member of a set (named as a domain) to another set of members (named as a codomain). This term has a different meaning from the same word that is used every day, such as "the tool works well." The concept of function is one of the basic concepts of mathematics and any quantitative science. The terms "function", "mapping", "map", "transformation" and "operator" are usually used synonymously.
In function, there are several important terms, including:
The domain is the area of origin of the function f denoted by Df.
Codomain is the area where the f function area is denoted by Kf.
The range is the result area which is a subset of the codomain. The function range f is denoted by Rf.
PROPERTIES OF FUNCTIONS
1. INJECTIVE FUNCTION
Called one-on-one function. Suppose the function f represents A to B, then the function f is called a one-on-one function (injective), if each two different elements in A will be mapped to two different elements in B. Furthermore, it can be said briefly that f: A → B is injective function if a ≠ b results in f (a) ≠ f (b) or equivalent if f (a) = f (b) then the effect is a = b.
2. SURJECTIVE FUNCTION
Function f: A → B is called a function to or objective function if and only if for any b in the B domain can be at least one an in domain A so f (a) = b applies. In other words, a codomain of the objective function is the same as its range.
3. BIJECTIVE FUNCTION
A mapping off: A → B is such that f is both an injective and objective function at once, so it is said "f is a function of wisdom" or "A and B are in one-to-one correspondence".
Therefore, The table 1 represent the function From the given figure.
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The average car sold from dealership a is $25,700.if the salesperson receives 1.5%commission on the price of the car ,how much commission is made on the average car sold
Answer: $385.50
Step-by-step explanation: 25,700x1.5% = 385.50
if log 5 a= 3, then log 5 (a^3) =
There a rule for logarithms:
[tex]\log_n(a^m) = m \cdot \log_n(a)[/tex]
So when you're working with [tex]\log_5(a^3)[/tex], we can use that rule to rewrite this as
[tex]\log_5(a^3) = 3 \cdot \log_5(a)[/tex]
And since the first part tells us (on paper) that [tex]\log_5(a) = 2[/tex], then we can say
[tex]\log_5(a^3) = 3 \cdot \log_5(a) = 3 \cdot 2 = 6[/tex]
The probability of rolling a number greater than 1 on a six sided number cube is 5/6 choose the likelihood that best describes the of this event. A neither likely or unlikely B impossible C Unlikely D Likely
The probability of rolling a number greater than 1 on a six sided number cube is 5/6 therefore the likelihood that best describes the of this event is that it is Likely and is therefore denoted as option D.
What is Probability?
This is referred to as the branch of mathematics which describes how likely an event is to occur or how likely it is that a proposition is true using numerical descriptions.
In this scenario we were told that the probability of rolling a number greater than 1 on a six sided number cube is 5/6 which means that is likely to occur due to the high percentage or ratio of the event being observed thereby making option D the correct choice.
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-
The gas mileages (in milles per gallon) for 35 cars are shown in the frequency distribution. Approximate the mean of the frequency
distribution
The approximate mean of the frequency distribution is
(Round to one decimal place as needed.)
Gas Mileage
(in miles per gallon)
27-30
31-34
35-38
39-42
Frequency
12
14
3
6
Answer:
The gas mileages (in miles per gallon) for 35 cars are shown in the frequency distribution. This means that the gas mileage of 35 cars has been recorded and grouped into categories. The categories are 27-30, 31-34, 35-38, and 39-42. The frequency is the number of cars in each category. For example, there are 12 cars that have a gas mileage between 27 and 30 miles per gallon. Approximating the mean of the frequency distribution means finding the average gas mileage of all the 35 cars. To do this, you need to add the gas mileages of all the 35 cars and divide the sum by 35. The approximate mean of the frequency distribution is 33.4 miles per gallon. An analogy that can help you understand this concept is that of a classroom full of students. If you want to calculate the average height of the students, you would need to add up the heights of all the students and divide the sum by the total number of students. The same concept applies to the frequency distribution.
GIVE ME BRAINIESTWrite a system of equations to describe the situation below, solve using any method, and fill in the blanks.
The postal service offers flat-rate shipping for priority mail in special boxes. Today, Jaden shipped 4 small boxes and 2 large boxes, which cost him $36 to ship. Meanwhile, Pamela shipped 3 small boxes and 9 large boxes, and paid $102. How much does it cost to ship these two sizes of box?
Shipping costs $
for a small box and $
for a large box.
The cost to ship the small boxes and large boxes are 4 and 10 dollars respectively.
How to represent system of equation?The postal service offers flat-rate shipping for priority mail in special boxes. Today, Jaden shipped 4 small boxes and 2 large boxes, which cost him $36 to ship. Meanwhile, Pamela shipped 3 small boxes and 9 large boxes, and paid $102.
The system of equation of represent the situation is as follows:
let
x = cost for small boxes
y = cost for large boxes
Hence,
4x + 2y = 36
3x + 9y = 102
Let's find how much used to ship the two sizes of box.
Multiply equation(i) by 4.5
18x + 9y = 162
3x + 9y = 102
Subtract the equations
15x = 60
x = 60 / 15
x = 4
Therefore,
2y = 36 - 4x
2y = 36 - 4(4)
2y = 36 - 16
2y = 20
y = 20 / 2
y = 10
Therefore,
cost for small boxes = 4 dollars
cost for large boxes = 10 dollars
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Find four consecutive even integers such that twice the sum of the second and third exceeds 3 times the first by 32.
Answer:
20, 22, 24, 26-------------------------------------
Let the integers be:
x, x + 2, x + 4, x + 6Twice the sum of the second and third exceeds 3 times the first by 32:
2(x + 2 + x + 4) = 3x + 32Solve it for x:
2(2x + 6) = 3x + 324x + 12 = 3x + 324x - 3x = 32 - 12x = 20consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. given that the volume of this set is $\displaystyle {{m n\pi}\over p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m n p$.
The solution of m+n+p = 462+40+3 = 505.
The set is divided into several parts: the large 3x4x5 parallelepiped, 6 external parallelepipeds that share a face with the large parallelepiped and have a height of 1, 1/8 spheres (one at each vertex of the large parallelepiped), and 1/4 cylinders connecting each adjacent pair of spheres.
The parallelepiped has a volume of $3 times 4 times 5 = 60 cubic units.
The external parallelepipeds have a volume of $2(3 times 4 times 1)+2(3 times 5 times 1)+2(4 times 5 times 1)=94$.
Each of the 1/8 spheres has a radius of 1. Their combined volume is [tex]\frac{4}{3} \pi[/tex].
There are 12 of the 1/4 cylinders, allowing for the formation of 3 complete cylinders. Their volumes are 3[tex]\pi[/tex], 4[tex]\pi[/tex], and 5[tex]\pi[/tex], totaling 12[tex]\pi[/tex].
The total volume of these parts is 60+94+ [tex]\frac{4}{3} \pi[/tex] +12[tex]\pi[/tex] = [tex]\frac{462 + 40\pi }{3}[/tex]. Thus, the solution is m+n+p = 462+40+3 = 505.
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Given f(x) = x - 7 and g(x) = x2.
Find g(f(4)).
g(f(4)) means to first find f(4) and then use that result as the input for g(x).
So,
f(4) = 4 - 7 = -3
then we use the result, -3 as the input for g(x)
g(-3) = (-3)^2 = 9
Therefore, g(f(4)) = 9
Larry leaves home at 9:06 and runs at constant speed to the lamppost 200 m away. He reaches the lamppost at 9:09, immediately turns, and runs to the tree (another 1,000 m away). Larry arrives at the tree at 9:13.
What is Larry's average velocity, in m/min, during each of these two intervals. Find v1 and v2.
Larry ran at an average velocity of 66.7 m/min to the lamppost 200 m away and an average velocity of 250 m/min to the tree 1000 m away.
v1 = 200 m / 3 min = 66.7 m/min
v2 = 1000 m / 4 min = 250 m/min
Larry left his home at 9:06 and ran at a constant speed to the lamppost 200 m away. He arrived at the lamppost at 9:09. Immediately after arriving, he turned around and ran to a tree 1000 m away, arriving at 9:13. To find his average velocity during each interval, we can divide the distance traveled by the time it took to travel the distance. For the first interval, Larry traveled 200 m in 3 minutes, so his average velocity was 200 m divided by 3 min, or 66.7 m/min. For the second interval, Larry traveled 1000 m in 4 minutes, so his average velocity was 1000 m divided by 4 min, or 250 m/min.
Larry ran at an average velocity of 66.7 m/min to the lamppost 200 m away and an average velocity of 250 m/min to the tree 1000 m away.
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1.02 quiz what is the minimum of the sinusoidal?
Answer:
The function's minimal value is m = A |B|. When either sin x or cos x is equal to 1, this minimum is reached. Create a graph for the first example, y = 1 + 2 sin x.
Step-by-step explanation:
what is an explicit formula for 9, 3, 1, 1/3
An explicit definition can give a value for the nth item in a sequence by simply using the value of n: an = -27 * (-1/3)n-1
What is Geometric Progression ?
Geometric Progression can be defined as the sequence of numbers in which the common ratio between any consecutive numbers is same.
When you say, "each term in the sequence is obtained by multiplying -1/3 to the previous term," that is the recursive definition.
An explicit definition can give a value for the nth item in a sequence by simply using the value of n:
an = -27 * (-1/3)n-1
(more generally written as: an = a1 * (-1/3)^n-1 )
if n = 1
we get,
an = -27 *1 = -27
if n=2
we get,
an = 9
and if n=3
we get,
an = -3
similarly if n=4
than an= 1
Therefore, An explicit definition can give a value for the nth item in a sequence by simply using the value of n: an = -27 * (-1/3)n-1
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Indicate the equation of the given line in standard form. The line containing the altitude to the hypotenuse of a right triangle whose vertices are P(-1, 1), Q(3, 5), and R(5, -5). Show calculations.
The linear function containing the altitude to the hypotenuse of a right triangle whose vertices are P(-1, 1), Q(3, 5), and R(5, -5) is given as follows:
y = -5x + 20.
How to define the linear functions?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.The hypotenuse is composed by segment QR, hence the points are:
Q(3,5) and R(5,-5).
The slope is given by the change in y divided by the change in x, hence:
m = (-5 - 5)/(5 - 3)
m = -5.
Hence:
y = -5x + b.
When x = 3, y = 5, hence the intercept b is obtained as follows:
5 = -15 + b
b = 20.
Hence the equation is:
y = -5x + 20.
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Solve the inequality, then identify the graph of the solution -3x-3 less than or equal to 6
The solution of the inequality -3x - 3 ≤ 6 is x ≥ -3 represented by (D).
What is an equation?An equation is an expression that shows the relationship between numbers and variables. Equations are either linear, quadratic, cubic and so on
Inequality is an expression that shows the non equal comparison between two or more numbers and variables.
Number line contains number placed at regular intervals on a straight line.
Given the inequality:
-3x - 3 ≤ 6
adding 3 to both sides:
-3x - 3 + 3 ≤ 6 + 3
-3x ≤ 9
Dividing by -3:
x ≥ -3
The solution is D. It a solid dark circle beginning from -3 to the right.
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7 - (3 + 41) + 6i
-10 + (6 - 5i) - 9i
Solve with explanation pls
The simplified form of the first expression is -37 + 6i and the second expression is -16 - 14i.
What is the arithmetic operations?
Arithmetic operations are basic mathematical operations that can be performed on numbers, such as addition, subtraction, multiplication, and division.
The first expression is 7 - (3 + 41) + 6i. To simplify this expression, we need to first simplify the parentheses. Inside the parentheses, we have 3 + 41 = 44. So the expression becomes:
7 - 44 + 6i
Next, we can simplify the arithmetic operations by combining like terms:
7 - 44 + 6i = -37 + 6i
The second expression is -10 + (6 - 5i) - 9i. Similar to the first expression, we need to simplify the parentheses first. Inside the parentheses, we have 6 - 5i = 6 - 5i. So the expression becomes:
-10 + 6 - 5i - 9i
Next, we can simplify the arithmetic operations by combining like terms:
-10 + 6 - 5i - 9i = -16 - 14i
Hence, the simplified form of the first expression is -37 + 6i and the second expression is -16 - 14i.
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The population (in millions) of Calcedonia as a function of time t (years) is P(1) = 55 . 20^005t. Determine the number of years it takes the population to double. (Give an exact answer. Use symbolic notation and fractions where needed.) years: Let g(t) = a2^kt. Determine the expression for g(t + 1/ķ) (for any positive constants a and k). (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the expression in terms of a, k, and t.
g(t+1/k)
Interpret the obtained results. - The population does not change significantly over time. - The population increases by 1/k after one year. - The population doubles after 1/k years - The population halves after 1/k years
Give the expression in terms of a, k, and t is g(t+1/k) = a2^(k(t+1/k)) = a2^kt * 2^(1/k)
The population doubles when P(t) = 2 * P(0), so we can set up the equation:
2 * 55 = 20^(5t)
Solving for t:
t = log(2)/(5 * log(20)) = log(2)/log(400)
The population doubles after approximately 0.173 years or 63.1 days.
g(t+1/k) = a2^(k(t+1/k)) = a2^kt * 2^(1/k)
Interpretation: g(t+1/k) is the population at time t+1/k, where a and k are positive constants. The population at t+1/k is equal to the population at time t multiplied by 2^(1/k). This means that if k is a large number, the population will change very little over time, if k is a small number, the population will change more quickly.
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what is bd?
a. 10
b 12
c 13
d 14
use the figure shown for items 2-3
The value of BD is 12cm.
What is Pythagoras' theorem?
The Pythagorean theorem, sometimes known as Pythagoras' theorem, is a basic relationship between a right triangle's three sides in Euclidean geometry. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
Here, we have
BE = ED (as E is the center of BD - property of a kite).
By Pythagoras' theorem:
10² = 8² + BE²
BE² = 10² - 8² = 36
BE = 6
So BD = 2*6 = 12.
Hence, the value of BD is 12cm.
To learn more about Pythagoras' theorem from the given link
https://brainly.com/question/16176867
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Which expression is equivalent to 20y² + 5y?
O 5y(4y - 1)
O-5y(4y + 1)
O 5y(4y + 1)
O-5y(4y - 1)
Answer:
[tex]5y\left(4y+1\right)[/tex]Step-by-step explanation:
[tex]20y^2+5y[/tex]
[tex]20yy+5y[/tex]
[tex]5\cdot \:4yy+5\cdot \:1\cdot \:y[/tex]
[tex]5y\left(4y+1\right)[/tex]
Step-by-step explanation:
5y(4y + 1)
= 20y × y + 5y × 1
= 20y² + 5y
NB: BODMAS
6[tex]6\sqrt{24} \alphax^{4}[/tex]