A function that represents the amount of air that is f(r)=4/3π[3r²+3r+1]
Since a function is a rule that applies a specific output to a specific input, we can define a function in a variety of ways. This could involve a change in variables, for example, or defining a function that represents a rate of change.
The amount or volume of air required to inflate the spherical balloon from a radius of r inches to a radius of r+1 inches.
The formula for the volume of a sphere is V = 4/3 πr³
where V = volume and r = radius.
V= 4/3π[(r+1)³−r³]
V=4/3π[r³+3r²+3r+1−r³]
V =4/3π[3r²+3r+1]
For r in inches and V(r) in inches cubed,
f(r)=4/3π[3r²+3r+1].
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functions that aren't invertible can be made invertible by restricting their domains. for example, the function $x^2$ is invertible if we restrict $x$ to the interval $[0,\infty)$, or to any subset of that interval. in that case, the inverse function is $\sqrt x$. (we could also restrict $x^2$ to the domain $(-\infty,0]$, in which case the inverse function would be $-\sqrt{x}$.) similarly, by restricting the domain of the function $f(x).
When the domain of x is restricted, the function x2 becomes one-to-one, and the inverse function is x.
It is true that limiting a function's domain might cause it to become invertible. Because it is not one-to-one in the example, the function x2 is not invertible across its complete domain of real numbers (i.e., it assigns multiple outputs to some inputs). However, x2 becomes one-to-one and invertible when the domain of x is constrained to or any subset of that interval. The range of x2, which is, is used to define the inverse function in this situation, which is
The function x2 becomes one-to-one when the domain of x is constrained, and the inverse function is
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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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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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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
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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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
A projectile is launched from ground level with an initial velocity of feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by . Find the time(s) that the projectile will (a) reach a height of 192 ft and (b) return to the ground when 112 feet per second.
The time(s) that the projectile will (a) reach a height of 192 ft is 4 seconds and (b) return to the ground when is 8 seconds
What is time?Time is the continued sequence of existence and event that can occur in an apparently inevitable succession from the past through the present
The set height, s=192
(a) 192=-16t²+112t
-16t^2+112t-192=0
-t^2+8t-16=0
t^2-8t+16=0
(t-4)(t-4)=0
t=4 twice
The projectile will reach a height of 192ft after 4 sec on the way up and after 4 sec again on the way down
..
Also, set height, s=0
(b) -16t^2+128t=0
-t^2+8t=0
-t(t-8)=0
t=0 (reject)
or
t=8
The projectile will return to the ground after 8 sec
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Correct question:
A projectile is launched from the ground with an initial velocity of V0 feet per second. Neglecting air resistance, its height in feet per seconds after launch is given by s = -16t^2 +V0t. Find the time(s) that the projectile will
(a) reach a height of 192ft and
(b) return to the ground when V0= 112 feet per second.
An airplane travels 4466 kilometers against the wind in 7 hours and 5866 kilometers with the wind in the same amount of time. What is the rate of the plane in still air and what is the rate of the wind? Note that the ALEKS graphing calculator can be used to make computations easier. Rate of the plane in still air: Rate of the wind: km h km h 010 S
The Rate of plane in still air and rate of wind is 738 , 100 respectively .
What is Linear equation ?
Linear equation can be defined as the equation in which the highest degree is one
Given ,
An airplane travels 4466 kilometers against the wind in 7 hours and 5866 kilometers with the wind in the same amount of time.
Let speed of plane be Vp and Va
Now,
Vp - Va = 4466 / 7
Vp+Va = 5866 / 7
So, Evaluating the equations,
we get,
2Vp = (4466+5866) / 7
Vp = 1476 / 2
Vp = 738
So substitute Vp =738 in Vp-Va = 4466/7
738 - Va = 638
Va = 738 - 638
Va = 100.
Hence, The Rate of plane in still air and rate of wind is 738 , 100 respectively .
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What is the domain of f(x) = 5x – 7?
{x | x > –7}
{x | x < –7}
{x | x > 0}
{x | x is a real number}
Answer:
{x | x is a real number}
Step-by-step explanation:
x can have any real number value.
i could really use some help please
Answer:
B 6
Step-by-step explanation:
Answer:
B. 6
Step-by-step explanation:
6 x -14 = 3x + 4
3x = 18
X = [tex]\frac{18}{3}[/tex] Which equals =
6
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]
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.
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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Cathy saves 4/5 of the money she earns from her part-time job. How much does she save when she makes $140?
Step-by-step explanation:
Money she makes=$140
since she saves 4/5 of the money she earns that means that: 4/5×140 = $32
This means that she saves $32 out of $140 she makes or earn
6[tex]6\sqrt{24} \alphax^{4}[/tex]
which of the following is most likely to contribute to a statistically significant but clinically meaningless value
When an occurrence is so common that it shouldn't be the result of chance or external influences on the study, it is said to be "statistically significant."
Different clinical significance exists. It alludes to tangible results. It frequently refers to the actual impact of a medical problem (or medical treatment) on a patient's health. The effects of a medicine are clinically meaningful if they provide alleviation. A medical problem is also considered clinically significant if it affects a patient's quality of life or expected lifespan.
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Answer the following
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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evaluate the limit by first interpreting the sum as a riemann sum for a function defined on [0, 16].
The limit in question is
[tex]$$\lim_{n\to \infty} \frac{1^3 + 2^3 + 3^3 + \dots + n^3}{n^4}$$[/tex]
This limit can be interpreted as the Riemann sum of a function defined on the interval [0, 16]. To calculate the sum, we need to first find the function that is being approximated. To do this, we can use the fact that the numerator of the limit can be rewritten as a single summation:
[tex]$$\sum_{k=1}^n k^3 = \frac{n^2 (n+1)^2}{4}$$[/tex]
This means that the function we are approximating is [tex]$f(x) = \frac{x^2 (x+1)^2}{4}$[/tex]
Now, to calculate the Riemann sum, we need to divide the interval [0, 16] into $n$ equal subintervals. The width of each subinterval is given by [tex]$\Delta x = \frac{16}{n}$\\The Riemann sum is given by $$\sum_{i=1}^n f\left(x_i\right) \Delta x = \sum_{i=1}^n \frac{\left(x_i\right)^2 \left(x_i+1\right)^2}{4} \cdot \frac{16}{n}$$[/tex]
where [tex]$x_i = \frac{16i}{n}$[/tex]
Plugging this into the limit, we get
[tex]$$\lim_{n\to \infty} \frac{1^3 + 2^3 + 3^3 + \dots + n^3}{n^4}\\ = \lim_{n\to \infty} \frac{\sum_{i=1}^n \frac{\left(x_i\right)^2 \left(x_i+1\right)^2}{4} \cdot \frac{16}{n}}{n^4} = \\frac{16}{4} \lim_{n\to \infty} \frac{\sum_{i=1}^n \frac{\left(x_i\right)^2 \left(x_i+1\right)^2}{n}}{n^3}$$[/tex]
Finally, we can take the limit of the above expression to get
[tex]$$\lim_{n\to \infty} \frac{1^3 + 2^3 + 3^3 + \dots + n^3}{n^4} = \frac{16}{4} \cdot \frac{f(16)}{16^3} = \frac{1}{4}$$[/tex]
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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:
consider 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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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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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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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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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 = 20Write 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
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
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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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.
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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.