Answer: Each person can get 12 cookies, or 12.2
Step-by-step explanation:
All you have to do is divide 61 by 5, your answer would 12 remainder 1, you would then divide 1 by 5 to get 0.2 . Add 12 and .2 to get your answer. The quotient would be 12 and the remainder 1.
Which equation below illustrates the associative property of addition?
(1) (2+8)+(1+9)=(8+2)+(9+1)
(3) 5+2(3+4)=5+6+8
(4) (3+7)+2=3+(7+2)
(2) 5(2+7)=10+35
Option number (4) i.e., (3+7)+2=3+(7+2) illustrates the associative property of addition.
As per the question statement, we are supposed to identify the equation which illustrates the associative property of addition.
Before solving this, we should be aware about the concept of associative property of addition which states that rearranging the parentheses in an expression will not change the result i.e., a+(b+c) = (a+b)+c
Therefore using the above property, we conclude that option number (4) i.e., (3+7)+2=3+(7+2) illustrates the associative property of addition.
As (3+7)+2=3+(7+2) can be written in the standard form a+(b+c)=(a+b)+c which is a characteristic equation of associative property of addition.
Associative property: Rearranging the parenthesis in an expression will not change the outcome since some binary operations have the associative feature.e.g., a+(b+c)=(a+b)+c OR a*(b*c)=(a*b)*c
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positive or negative ?
Answer:The answer would be negative.
Step-by-step explanation:There is already a negative balance of -$25 and you would add $23.-25+23=-2.So therefore the answer would be negative.
Please give brainliest if answer is correct.
The temperature outside was -17 degrees. It climbed 9 degrees during the day beore dropping 10 degrees at night. What was the temperature at night?
The temperature outside at night is 2 degrees.
One degree Celsius, for instance, is one-hundredth of the temperature difference between the point at which water changes state from solid to liquid to its gaseous stage. A degree can be defined as a fixed change in temperature measured against a specific scale.
Let's say the temperature outside is x.
Now, the initial temperature was - 17 degrees, then it rises to 9 degrees during the day before it drops 10 degrees at night.
So,
x = - 17 degrees
The temperature rises by 9 degrees.
Then,
x = - 17 degrees + 9 degrees
x = - 8 degrees
The temperature drops 10 degrees at night.
Then,
x = -8 degrees + 10 degrees
x = 2 degrees
Hence, the temperature at night is 2 degrees.
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Evaluate the following integral
[tex]\begin{gathered} \sf \int \: sec(x - a) \: sec(x - b) \: dx \\ \end{gathered}[/tex]
HELP I REQUEST !!
Answer:
[tex]\dfrac{\ln | \sec (x-b)- \ln | \sec (x-a)}{\sin (a-b)}+\text{C}[/tex]
Step-by-step explanation:
Fundamental Theorem of Calculus
[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]
If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.
Given indefinite integral:
[tex]\displaystyle \int \sec(x-a) \sec (x-b)\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Trigonometric Identity}\\\\$\sec \theta=\dfrac{1}{\cos \theta}$\\\end{minipage}}[/tex]
Therefore:
[tex]\implies \displaystyle \int \dfrac{1}{\cos(x-a)} \cdot \dfrac{1}{\cos (x-b)}\:\text{d}x[/tex]
[tex]\implies \displaystyle \int \dfrac{1}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]
[tex]\textsf{Multiply the integral by }\dfrac{\sin (a-b)}{\sin (a-b)}:[/tex]
[tex]\implies \displaystyle \int \dfrac{1}{\cos(x-a)\cos (x-b)} \cdot \dfrac{\sin (a-b)}{\sin (a-b)}\:\text{d}x[/tex]
Take the constant outside the integral:
[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin (a-b)}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]
Rewrite the numerator:
[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin [(x-b)-(x-a)]}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{6 cm}\underline{Trigonometric Identity}\\\\$\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B$\\\end{minipage}}[/tex]
Therefore:
[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin(x-b) \cos (x-a)-\cos (x-b) \sin (x-a)}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]
[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin(x-b) \cos (x-a)}{\cos(x-a)\cos (x-b)} -\dfrac{\cos (x-b) \sin (x-a)}{{\cos(x-a)\cos (x-b)}}\:\text{d}x[/tex]
[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin(x-b)}{\cos (x-b)} -\dfrac{\sin (x-a)}{\cos(x-a)}\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Trigonometric Identity}\\\\$\tan \theta=\dfrac{\sin \theta}{\cos \theta}$\\\end{minipage}}[/tex]
Therefore:
[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \tan(x-b)-\tan(x-a)\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{4.3 cm}\underline{Integrating $\tan x$}\\\\$\displaystyle \int \tan x\:\text{d}x=\ln | \sec x|+\text{C}$\end{minipage}}[/tex]
Therefore:
[tex]\implies \dfrac{1}{\sin (a-b)}\left[ \ln | \sec (x-b)- \ln | \sec (x-a)\right]+\text{C}[/tex]
[tex]\implies \dfrac{\ln | \sec (x-b)- \ln | \sec (x-a)}{\sin (a-b)}+\text{C}[/tex]
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How is "Mathematical Reality" different than physical reality?
Math reality is more restrictive and logical, physical reality is more "free" and well-behaved.
How is "Mathematical Reality" different than physical reality?Mathematics are a logical construct, thus, everything in the "mathematical realty" must follow a certain logic.
For example, in math, always that you do a simplification (like rounding, applying a theorem, using an integration property, etc) you need to prove logically why you can do that.
While on physics we assume the reality is "nice" and we can always apply the simplifications. This is because most of the functions that represent physics are nice (continuous, differentiable, etc) functions, in the same way, most of the matrices are square matrices, and so on.
Concluding, for example in math the number 4.99999 is exactly 4.9999
On physics, if that same number represents a measure, for example:
4.99999 meters is practically equal to 5 meters.
Math reality is more restrictive and logical, physical reality is more "free" and well-behaved.
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For a show during the half-time break of a football game, a stage will be set up on the field. The center of the stage is on the 50-yard line. If the distance from the center to the outer edges is 14 yards, which equation can be used to find the locations of the outer edges of the stage?
|x - 50| = 14 is the equation used to find the locations of the outer edges of the stage.
Absolute value function
The absolute function is defined by
|x| = x, x≥0
|x| = -x, x<0
It measures the distance of a point x to the origin, let's take some example
|-8| = 8
|-5| = 5
Here it is given that the center of the stage is on the 50-yard line, and the distance from the center to the outer edges is 14 yards.
So from this, we get an equation,
|x - 35| = 11
So the solution is given as follows:
First edge:
x - 50 = -14
x = 36 yards line
Second edge:
x- 50 = 14
x = 64 yard line
Therefore the equation used to find the location of the outer edges of the stage is |x - 50| = 14.
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For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.
f(x)= √-2x+3
If the inverse of f exists, it is represented by f⁻¹ and exists only if f is a bijective function.
The inverse of the function is f⁻¹ = - (x² - 3/2)
Domain is 3/2 to positive infinity
Range is 0 to positive infinity.
What exactly is an inverse function?The inverse function of a function f is a function that reverses the operation of f. The inverse of f exists if and only if f is bijective, and it is denoted by f⁻¹ if it exists.
A function's inverse is not always a function. To ensure that the inverse function is also a function, the original function must be a one-to-one function. A one-to-one function is one in which each second element corresponds to exactly one first element.
Let the given function be f(x) = √-2x+3
y = √-2x+3
simplifying the value of x, we get
x = √-2x+3
x² = -2y + 3
simplifying the above equation, we get
x² - 3 = -2y
y = - (x² - 3/2)
Therefore, the inverse function be f⁻¹ = - (x² - 3/2)
Domain is 3/2 to positive infinity because a negative number cannot be square-rooted.
Range is 0 to positive infinity.
f⁻¹ = - (x² - 3/2)
The domain of the inverse is negative infinity to positive infinity, which is not a mirror image of the range of the original equation.
Anything squared is a positive number or zero.
The minimum value of the range is
f⁻¹ = - (x² - 3/2)
f⁻¹ = -(-3/2)
f⁻¹ = 3/2
The range of the inverse is 3/2 to positive infinity, a mirror of the domain of the original equation.
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The altitude of an equilateral triangle is 18 inches. Find the length of a side. Sketch the figure. Round decimals to the nearest tenth (1 decimal place).
side length =
The length of the side of the equilateral triangle is 20.76 inches.
Here it is given that the length of the altitude of the equilateral triangle is 18 inches.
The formula of the altitude(h) of the equilateral triangle is
h = [tex]\frac{\sqrt{3} }{2}[/tex]a
Here a is the side of an equilateral triangle.
So we have
18 = [tex]\frac{\sqrt{3} }{2}[/tex]a
a = 18 × 2/[tex]\sqrt{3}[/tex]
For the rationalization of the fraction multiply numerator and denominator by [tex]\sqrt{3}[/tex].
a = [tex]\frac{18 * 2 * \sqrt{3} }{\sqrt{3}\sqrt{3} }[/tex]
= [tex]\frac{18 * 2 * \sqrt{3} }{3}[/tex]
= 6 × 2 × √3
= 12×√3
= 12 × 1.73
= 20.76 inches
Therefore the length of the side of the triangle is 20.76 inches.
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A student population of 1200 was estimated to increase by 15% in the next five years. the population actually increased by 20%. find the most estimated and actual student populations and describe the percent error
If a student population of 1200 was estimated to increase by 15% in the next five years but the population actually increased by 20%, then the most estimated and actual student populations are equal to 1380 and 1440 students respectively, and the percent error is equal to 4.2%.
The estimated and actual student populations can be calculated as follows,
Estimated increase = 15% = 15 / 100 = 0.15
Initial population = 1200
0.15 × 1200 = 180
Estimated population in the next five years = 1200 + 180 = 1380
Actual increase = 20% = 20 / 100 = 0.2
Initial population = 1200
0.2 × 1200 = 240
Actual population in the next five years = 1200 + 240 = 1440
We can find the percentage error as follows,
percentage error = [(actual value - estimated value) / actual value] × 100
percentage error = [(1440 - 1380) / 1440] × 100
percentage error = [60 / 1440] × 100
percentage error = 0.042 × 100 = 4.2%
Hence the estimated and actual student populations is calculated to be 1380 and 1440 respectively and the percentage error is equal to 4.2%.
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Write an equation that equals 2,456
Answer:
2456x =
Step-by-step explanation:
Simple i know but you didnt really specify.
Answer: 4,912/2=2,456
Step-by-step explanation:hope it helps, brainliest? i could really use it
HALP I NEED HALP AND I DONT GOT A LOT POINTS LEFT PLS I NEED THIS.
Its just vector sums pls help
i dont get this and need to submit by tonight PLEASE HELPP THANKKS
Based on the dimensions of the shape, the length of PR can be found to be 19.8 units.
What is the length of PR?The given triangle is an equilateral triangle which means that all the sides are equal.
PR = QR
This means that:
2n + 9 = 7n - 18
Solving for n gives:
7n - 2n = 18 + 9
5n = 27
n = 5.4
The length of PR is therefore:
= 2n + 9
= 2 (5.4) + 9
= 19.8 units
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A boat is carrying containers that weigh 4000 pounds each.
Use this information to fill in the table. Then plot the ordered pairs given by the table.
By applying direct proportion, the information should be filled in the table are as follows:
Number of containers Weight (in pounds)
4 16,000
8 32,000
10 40,000
What is a proportion?A proportion can be defined as an equation which is typically used to represent (indicate) the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.
Mathematically, a direct proportion can be represented the following equation:
y = kx
Where:
y and x are the variables.k represents the constant of proportionality.Since the boat carried containers that weigh 4000 pounds each, we would multiply each of containers by 4000 as follows:
Number of containers Weight (in pounds)
4 16,000
8 32,000
10 40,000
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AT lunchtime, benjamin often borrows money from his friends to buy snacks in the school cafeteria benjamin borrowed $0.75 from his friend cltyde five days last week to buy ice cream bars represent the amount ben borrowed as the product of the two rational numbers; then, determine how much benjamin owed his friend last week
The total amount that's is borrowed by Benjamin is $3.75.
How to calculate the valueIt should be noted that from the information, Benjamin borrowed $0.75 from his friend cltyde five days last week to buy ice cream.
Therefore, the product will be:
= Amount borrowed × Number of times
= $0.75 × 5
= $3.75
The amount borrowed is $3.75.
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Twice a number subtracted from 15 is 7
Answer:
4
Step-by-step explanation:
15-2x=7
8=2x
4=x
---
HOPE THIS HELPS <3
Answer:
x=11
Step-by-step explanation:
let us name the unknown number: x
15 - 2x = 7
collect like terms,
-2x= 7 - 15
-2x= - 8
divide both sides by -2( coefficient of x)
-2x = -8
__ __
-2 -2
x= 4
The sum of two numbers is 37. One of the numbers is 15 more than the other number. What are the two numbers?
Answer: 11 and 26
Step-by-step explanation:
Simplify
6-√√25
4
DONE
11
Answer:
1/4, alternate forms are .25 and 2^-2
Step-by-step explanation:
Square Root of 25 is 5
6-5 =1
1/4
Aight my brother needs some help he is like 10 and doesn't know the answer and I can't bother. please help him.
[27 x 5 -(92 : 4 -36:3) x 5]:5
Answer:[tex]\frac{-11x^2+135}{5}[/tex]
Step-by-step explanation:
someone help me with thisss
The equation of the line that is perpendicular to the given line and that passes through the given points is x + 3y = 6
From the question, we are to determine the equation of the described line.
First,
We will determine the slope of the line that the equation was given.
The equation of the line is
y = 3x - 9
Comparing the equation to the general form of the equation of a straight line
y = mx + b
Where m is the slope
and b is the y-intercept
Thus,
By comparison,
m = 3
That is, the slope of the given line is 3
NOTE: Two lines are perpendicular if the product of their slope is -1.
That is,
m₁ × m₂ = -1
3 × m₂ = -1
m₂ = -1/3
The slope of the line that is perpendicular to the given line is -1/3.
We have that the line passes through the point (3, 1).
Then,
Using the point-slope form of the equation of a line
y - y₁ = m(x - x₁)
m = -1/3
x₁ = 3
y₁ = 1
Then,
y - 1 = -1/3(x - 3)
Multiply through by 3
3(y -1) = -1(x - 3)
3y - 3 = -x + 3
3y + x = 3 + 3
x + 3y = 6
Hence, the equation of the line that is perpendicular to the given line and that passes through the given point is x + 3y = 6
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Which expression is equivalent to (2)−3(4)0 ?
A = (12)×(12)×(12)×4
B= (12)×(12)×(12)×1
C= (−2)×(−2)×(−2)×1
D= (−2)×(−2)×(−2)×4
Answer:
if any number power of 0 is 1
so, (4)^0 =1
if any number power of -1 is equal to one divide this number.
like;
a^(-1) = 1/a
Therefore,
2^(-3) = 2^(-1) x 2^(-1) x 2^(-1)
= 1/2 x 1/2 x 1/2
Hence solution of 2^(-3). (4) ^0 is,
1/2 x 1/2 x 1/2 x 1
A survey of 62 customers was taken at a bookstore regarding the types of books purchased. The survey found that 38 customers
purchased mysteries, 31 purchased science fiction, 21 purchased romance novels, 18 purchased mysteries and science fiction, 12
purchased mysteries and romance novels, 9 purchased science fiction and romance novels, and 5 purchased all three types of
books.
a) How many of the customers surveyed purchased only science fiction?
b) How many purchased mysteries and science fiction, but not romance novels?
c) How many purchased mysteries or science fiction?
d) How many purchased mysteries or science fiction, but not romance novels?
e) How many purchased exactly two types of books?
a) There were customers who purchased only science fiction.
(Simplify your answer.)
For given survey of 62 customers was taken at a bookstore regarding the types of books purchased,
a) The number of customers surveyed who purchased only science fiction = 9
b) The number of customers surveyed who purchased mysteries and science fiction, but not romance novels = 13
c) The number of customers surveyed who purchased mysteries or science fiction = 51
d) The number of customers surveyed who purchased mysteries or science fiction, but not romance novels = 46
e) The number of customers surveyed who purchased exactly two types of books = 2
In this question,
we have been given a survey of 62 customers was taken at a bookstore regarding the types of books purchased.
38 customers purchased mysteries,
31 purchased science fiction,
21 purchased romance novels,
18 purchased mysteries and science fiction,
12 purchased mysteries and romance novels,
9 purchased science fiction and romance novels,
and 5 purchased all three types of books.
Let S represents the set of customers who purchased science fiction books,
R represents the set of customers who purchased romance novels books and
M represents the set of customers who purchased mysteries books.
From given information,
n(S) = 31
n(R) = 21
n(M) = 38
n(M ∩ S) = 18
n(M ∩ R) = 12
n(S ∩ R) = 9
n(M ∩ R ∩ S) = 5
a) we need to find the number of customers surveyed who purchased only science fiction.
= n(S) - n(M ∩ S) - n(S ∩ R) - n(M ∩ R ∩ S)
= 31 - (18 - 5) - (9 - 5) - 5
= 31 - 13 - 4 - 5
= 9
b) we need to find the number of customers surveyed who purchased mysteries and science fiction, but not romance novels
= n(M ∩ S) - n(M ∩ R ∩ S)
= 18 - 5
= 13
c) we need to find the number of customers surveyed who purchased mysteries or science fiction
n(M ∪ S) = n(M) + n(S) - n(M ∩ S)
= 38 + 31 - 18
= 51
d) we need to find the number of customers surveyed who purchased mysteries or science fiction, but not romance novels
= n(M ∪ S) - n(M ∩ R ∩ S)
= 51 - 5
= 46
e) we need to find the number of customers surveyed who purchased exactly two types of books
= n(M ∩ S) - n(S ∩ R) - n(M ∩ R)
= (18 - 5) - (9 - 5) - (12 - 5)
= 13 - 4 - 7
= 2
Therefore, for given survey of 62 customers was taken at a bookstore regarding the types of books purchased,
a) The number of customers surveyed who purchased only science fiction = 9
b) The number of customers surveyed who purchased mysteries and science fiction, but not romance novels = 13
c) The number of customers surveyed who purchased mysteries or science fiction = 51
d) The number of customers surveyed who purchased mysteries or science fiction, but not romance novels = 46
e) The number of customers surveyed who purchased exactly two types of books = 2
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solve each inequality ||2x-1|-2|>3
Answer:
Solution: x < -2 or x > 3
Interval notation: (-∞, -2) ∪ (3, ∞)
Step-by-step explanation:
Given inequality:
[tex]||2x-1|-2| > 3[/tex]
Apply the absolute rule:
[tex]\textsf{If $|u| > a$, $a > 0$ \;then \;$u > a$ \;or \;$u < -a$}.[/tex]
Therefore:
[tex]\textsf{Case 1}: \quad |2x-1|-2 > 3[/tex]
[tex]\textsf{Case 2}: \quad |2x-1|-2 < -3[/tex]
Solve each case independently.
Case 1Isolate the absolute value on one side of the equation:
[tex]\begin{aligned}\implies |2x-1|-2& > 3\\\implies |2x-1| & > 5 \end{aligned}[/tex]
Apply the absolute rule:
[tex]\textsf{If $|u| > a$, $a > 0$ \;then \;$u > a$ \;or \;$u < -a$}.[/tex]
[tex]\begin{aligned}\underline{\textsf{Equation 1}} & & \quad \quad\underline{\textsf{Equation 2}}\\2x-1 & > 5 & 2x-1 & < -5\\2x& > 6 & 2x& < -4\\x& > 3 & x& < -2\end{aligned}[/tex]
Therefore, x < -2 or x > 3.
Case 2Isolate the absolute value on one side of the equation:
[tex]\begin{aligned}\implies |2x-1|-2& < -3\\\implies |2x-1| & < -1 \end{aligned}[/tex]
As an absolute value cannot be less than zero, there is no solution for x∈R.
SolutionSolution: x < -2 or x > 3
Interval notation: (-∞, -2) ∪ (3, ∞)
36x² - 65x + 25 = 0
Work out the value of x.
Let’s solve this step-by-step
First factor the equation,
(9x - 5) (4x - 5) = 0
Then set each factor equal to zero,
9x - 5 = 0 and 4x - 5 = 0
Both x = 5 / 9 or x = 5 / 4 are correct
Under her cell phone plan, Jaya pays a flat cost of $60.50 per month and $5 per gigabyte. She wants to keep her bill under $75 per month. Which inequality can be used to determine xx, the maximum number of gigabytes Jaya can use while staying within her budget?
The inequality used to determine Jaya's maximum number of Gigabytes available for use, within her budget is 60.50 + 5x ≤ 75.
What is inequality?
The relation between two expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than’, or < ‘less than’.
Given:
Jaya pays a flat cost of $60.50 per month and $5 per gigabyte and keep her bill under $75 per month.
According to given question we have
Let the cost of gigabyte be x
flat cost =$60.50 per month
Budget= $75 per month
i.e
60.50 + 5x ≤ 75
Therefore, the inequality used to determine Jaya's maximum number of Gigabytes available for use, within her budget is 60.50 + 5x ≤ 75.
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X² - x -12 ÷ x-4
3x-9. 12
The result of division of (x² - x - 12) by (x - 4) is (x + 3).
What is termed as factorization?The breaking or dissolution of an entity (for example, a number, a matrix, or even a polynomial) into such a product of some other entity, or factors, that when multiplied together give the previous figure or matrix, etc.It is simply dividing an integer or linear function into factors that, when multiplied together, result in the original or initial integer or polynomial. We use the factorisation method to simplify any algebraic as well as quadratic equation by representing it as the product of factors rather than expanding the brackets.The given equation is;
(x² - x - 12) / (x - 4)
where, (x² - x - 12) is the numerator and (x - 4) is the denominator.
Find the factors of the numerator.
= (x - 4)(x + 3)/(x - 4)
Cancel the values (x - 4) from the numerator an denominator part.
= (x + 3)
Thus, the solution of the factorization is found as (x + 3).
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Helppppp ASAP i’ll mark you the brainliest
Answer:
h = 0.01g
Step-by-step explanation:
[tex]{ \tt{g = kh + c}}[/tex]
When g is 0, h is 0 hence c is 0
[tex]{ \tt{g = kh}}[/tex]
when g is 1, h is 0.01
[tex]{ \tt{1 = 0.01k}} \\ { \tt{k = 100}} \\ { \boxed{ \tt{ equation \to \: {g = 100h}}}} \\ { \rm{or}} \\ { \boxed{ \tt{equation \to \: h = 0.01g}}}[/tex]
Carl writes 14/3 x 7/3 to find the quotient of 4 and 2/3 divided by 2 1/3 what is his mistake
Carl made a mistake by multiplying the two values in the quotient rather than dividing them.
What are quotients?In this example, the number divided by (15) is known as the dividend, and the number divided by (3 in this instance) is known as the divisor. The quotient is the outcome of the division.So, the result of 4 divided by 2 1/3 and an explanation of Carl's error:
4 1/3 is equal to 14/3.Additionally, 2 1/3 is equal to 7/3.Thus, if we divide 4 2/3 by 2 1/3, we get:
(14/3)/(7/3) = 14/3 × 3/7 = 14/7 = 2Carl made the error of assuming his quotient (2) was equal to the value of the terms it contained.
In actuality, 2 is just a multiplier for 7/3 and has no intrinsic value.Therefore, Carl made a mistake by multiplying the two values in the quotient rather than dividing them.
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Solve each equation. Check your answers.
log x+4=8
log (x + 4) = 8 => x = [tex]10^8[/tex], by properties of logarithm.
What is Logarithm?The opposite of exponentiation is the logarithm.This indicates that the exponent to which a fixed number, base b, must be raised in order to obtain a specific number x, is represented by the logarithm of that number.A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.Given:
log ( x + 4) = 8
Raising both sides by 10, in order to remove the logarithm.
=> [tex]10^{log_{10}(x+4)} = 10^{8}[/tex]
=> x + 4 = [tex]10^8[/tex] (as [tex]a^{log_a(x)} = x[/tex] )
=> x = [tex]10^8[/tex] (as the value of [tex]10^8[/tex] is so large, subtracting 4 from it won't make much difference).
Hence, log (x + 4) = 8 => x = [tex]10^8[/tex], by properties of logarithm.
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A 54-inch board is to be cut into three pieces so that the second piece is 3 times as long as the first piece and the third piece is 5 times as long as the first piece. If x represents the length of the first piece, find the length of all 3 pieces
Answer: The length of First piece = 6
The length of second piece = 18
The length of third piece = 30
Step-by-step explanation:
Given data,
A 54-inch board is to be cut into three pieces.
so that the second piece is 3 times as long as the first piece and the third piece is 5 times as long as the first piece.
So, we can write,
Let us assume, first piece is represented by = x
Then,
second piece is 3 times as long as the first piece
So, we can write,
second piece is represented by = 3 ( first piece )
second piece is represented by = 3x
Then,
third piece is 5 times as long as the first piece
So, we can write,
third piece is represented by = 5 ( first piece )
third piece is represented by = 5x
So, we can find the all 3 pieces length,
we can solve it :
combine all three pieces = x + 3x + 5x
length of all three pieces = x + 8x
= 9x
Total board is to be cut into three pieces = 54
Hence,
9 x = 54
x = 54/9
x = 6
Now we know that the base length, x, is equal to 6.
From there, we can find the length of the all three pieces is :
The length of First piece = x = 6
The length of second piece = 3x
= 3(6)
= 18
The length of third piece = 5x
= 5(6)
= 30
Therefore,
The length of First piece = 6
The length of second piece = 18
The length of third piece = 30
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Identify each function or situation as an example of exponential growth or decay. What is the y -intercept?
b. y=11(0.75x)
The given function is exponential growth and the y-intercept is y=11.
Given that the given function is y=11(0.75)ˣ.
Exponential growth or decay describes the process of reducing or growing an amount by a consistent percentage rate over a period of time and it is represented by the function y=a(1-b)ˣ where, y is the final amount, a is the original amount and b is the growth or decay factor and x is the amount of time passed.
If a is positive and b is greater than 1 then it is exponential growth.
If a is positive and b is less than 1 but greater than 0 then it exponential decay.
The graph of the given function y=11(0.75)ˣ is shown in attached image.
From the graph, it is visible that the given function is an exponential growth because a is 11 and it is positive and it is greater than 1.
Now, we will find the y-intercept by substituting x=0, we get
y=11(0.75)⁰
y=11(1)
y=1
Hence, the given function or situation y=11(0.75)ˣ as an example of exponential growth and the y-intercept is 11.
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