Answer:
Therefore, the correct option is: (C.) [tex]\( f(x) = 2 \cdot 4^x \)[/tex]
Explanation:
To determine the exponential function that passes through the given points, we can use the general form of an exponential function:
[tex]\[ f(x) = a \cdot b^x \][/tex]
Where:
[tex]\( a \)[/tex] is the initial value or the value of the function when [tex]\( x = 0 \).[/tex]
[tex]\( b \)[/tex] is the base of the exponential function.
Given the points [tex]\((1, 8)\) and \((4, 512)\)[/tex], we can substitute these coordinates into the general form and solve for [tex]\( a \) and \( b \).[/tex]
Using the point [tex]\((1, 8)\):[/tex]
[tex]\[ 8 = a \cdot b^1 \][/tex]
[tex]\[ 8 = a \cdot b \][/tex]
Using the point [tex]\((4, 512)\):[/tex]
[tex]\[ 512 = a \cdot b^4 \][/tex]
We can divide the equation obtained from the second point by the equation obtained from the first point to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \):[/tex]
[tex]\[ \frac{512}{8} = \frac{a \cdot b^4}{a \cdot b} \][/tex]
[tex]\[ 64 = b^3 \][/tex]
Taking the cube root of both sides:
[tex]\[ b = 4 \][/tex]
Now that we have found the value of [tex]\( b \)[/tex], we can substitute it back into one of the equations to find \( a \). Let's use the first point [tex]\((1, 8)\):[/tex]
[tex]\[ 8 = a \cdot 4 \][/tex]
[tex]\[ a = 2 \][/tex]
So, the exponential function that passes through the points [tex]\((1, 8)\) and \((4, 512)\)[/tex] is:
[tex]\[ f(x) = 2 \cdot 4^x \][/tex]
What is the measurements of ENV
Which characteristic is used to group artworks into periods or styles?
A: Abstract
B: Similar
The equation 9.95 + 0.30s=c gives the cost c in dollars that a website charges for downloading songs. The variable s stands for the number of songs downloaded. Find the cost of downloading 35 songs
9.95 + 0.30s=c
replaces s with 35
9.95 +0.30(35) =c
9.95 + 10.50 = 20.45
it will cost $20.45
Branliest offered
A. What is the circumference of the colony?
B. What is the radius of the colony?
Bacteria lives in groups called colonies colonies are usually circular the diameter of a particular bacterial colony is 12 mm their circumference of a circle is equal to Pi 3.14 times its diameter c= πd
A charity organization had a fundraiser where each ticket was sold for a fixed price. They had to sell a few tickets just to cover necessary production costs of $1200\$1200 $1200 dollar sign, 1200 . After selling 200200 200 200 tickets, they had a net profit of $12,000\$12{,}000 $12,000 dollar sign, 12, comma, 000 . Let P(n)P(n) P(n) P, left parenthesis, n, right parenthesis denote the net profit from the fundraiser PP P P (measured in dollars) as a function of the number of tickets sold nn n n . Write the function's formula
Answer:
[tex]P(n)=66n-1200[/tex]
Step-by-step explanation:
Let x be the revenue collected from selling each ticket.
We have been given that a charity organisation had to sell a few tickets just to cover necessary production costs of $1200. After selling 200 tickets, they had a net profit of $12,000.
Since we know that net profit is the difference between total revenue and total cost.
[tex]\text{Net profit}=\text{Total revenue- Cost}[/tex]
We can represent our given information in an equation to find the revenue collected from each ticket:
[tex]12000=200x-1200[/tex]
Let us solve for x by adding 1200 to both sides of equation.
[tex]12000+1200=200x[/tex]
[tex]13200=200x[/tex]
[tex]x=\frac{13200}{200}[/tex]
[tex]x=66[/tex]
Therefore, the revenue collected from selling each ticket is $66.
Now let us write the net profit, P(n), from fundraiser as the function of number of tickets sold,n.
The revenue from selling n tickets will be 66n.
Production costs = 1200
[tex]P(n)=66n-1200[/tex]
Therefore, our desired function will be: [tex]P(n)=66n-1200[/tex].
Color in equations what to color in 1+2+3+4=10 What cubes to color in?
Find the indefinite integral of [tex] \int\limits {\frac{5}{x^\frac{1}{2}+x^\frac{3}{2}} \, dx [/tex]
I have been able to simplify it to [tex] \int\limits {\frac{5\sqrt{x}}{x^3+x}} \, dx [/tex] but that is confusing,
I then did u-subsitution where [tex]u=\sqrt{x}[/tex] to obtain [tex] \int\limits {\frac{5u}{u^6+u^2}} \, dx [/tex] which simplified to [tex] \int\limits {\frac{5}{u^5+u}} \, dx [/tex], a much nicer looking integrand
however, I am still stuck
ples help
show all work or be reported
Answer:
[tex] \displaystyle10 \tan^{-1}( \sqrt {x}^{ } ) + \rm C[/tex]
Step-by-step explanation:
we would like to integrate the following integration:
[tex] \displaystyle \int \frac{5}{ {x}^{ \frac{1}{2} } + {x}^{ \frac{3}{2} } } dx[/tex]
in order to do so we can consider using u-substitution
let our
[tex] \displaystyle u = {x}^{ \frac{1}{2} } \quad \text{and} \quad du = \frac{ {x}^{ - \frac{1}{2} } }{2} [/tex]
to apply substitution we need a little bit arrangement
multiply both Integral and integrand by 2 and ½
[tex]\displaystyle 2\int \frac{1}{2} \cdot\frac{5}{ {x}^{ \frac{1}{2} } + {x}^{ \frac{3}{2} } } dx[/tex]
factor out [tex]x^{\frac{1}{2}}[/tex]:
[tex]\displaystyle 2\int \frac{1}{2 {x}^{ \frac{1}{2} } } \cdot\frac{5}{ (1+ {x}^{ } )} dx[/tex]
recall law of exponent:
[tex]\displaystyle 2\int \frac{ {x}^{ - \frac{1}{2} } }{2 } \cdot\frac{5}{ (1+ {x}^{ } )} dx[/tex]
apply substitution:
[tex]\displaystyle 2\int \frac{5}{ 1+ {x}^{ } } du[/tex]
rewrite x as [tex]\big(x^{\frac{1}{2}}\big)^{2}[/tex]:
[tex]\displaystyle 2\int \frac{5}{ 1+ ( {x ^{ \frac{1}{2} } })^{ 2 } } du[/tex]
substitute:
[tex]\displaystyle 2\int \frac{5}{ 1+ ( u)^{ 2 } } du[/tex]
recall integration rule of inverse trig:
[tex] \displaystyle 2 \times 5 \tan^{-1}(u)[/tex]
simplify multiplication:
[tex] \displaystyle10 \tan^{-1}(u)[/tex]
substitute back:
[tex] \displaystyle10 \tan^{-1}( {x}^{ \frac{1}{2} } )[/tex]
simplify if needed:
[tex] \displaystyle10 \tan^{-1}( \sqrt{x} )[/tex]
finally we of course have to add constant of integration:
[tex] \displaystyle10 \tan^{-1}( \sqrt {x}^{ } ) + \rm C[/tex]
and we are done!
Brand A cereal contains 10 cups of cereal. How many more cups of cereal are in Brand B cereal? 30% more cereal
Answer:
there are 13 more cups
Step-by-step explanation:
i already did it
In 1997, there were 43.2 million people who used free weights. Assuming the use of free weights increases 6% annually, which equation can be used to predict the number of people using free weights t years from 1997?
p = 43.2(0.06)t
p = 43.2(1.06)t
p = 43.2(0.94)t
p = 43.2(1.005)t
The correct equation to predict the number of free weight users is option B) p = 43.2(1.06)t. This equation accounts for a 6% annual increase in users.
Predicting the Number of Free Weight Users
In 1997, there were 43.2 million people using free weights. Given that the annual increase rate is 6%, we use the exponential growth formula to predict the number of people in future years. The correct equation to use is:
p = 43.2(1.06)t
This equation represents an exponential increase, where 'p' is the number of people using free weights and 't' is the number of years since 1997. This can be explained by the fact that a 6% annual increase multiplies the original number of users by 1.06 each year.
Which postulate or theorem can be used to prove that △ABC≅△ADC? ASA Congruence Postulate SAS Congruence Postulate HL Congruence Theorem SSS Congruence Postulate
Both the triangles ΔABC and ΔADC are congruent by SSS postulate of congruence.
From the picture attached,
In ΔABC and ΔADC,
AB ≅ AD (Given)
BC ≅ CD (Given)
AC ≅ AC (Reflexive property)
Since, corresponding sides of ΔABC are equal to ΔADC,
Therefore, ΔABC ≅ ΔADC (By S-S-S postulate of congruence)
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Helppp 3 out of the 5 im gunna post!
look where the line is on the y axis at x =2
answer is y=-3
value is -3
What is 24 ten thousands in standard form
We have that the standard form of 24 ten thousands is
240,000
From the Question we are told that
24 ten thousands is given
Where
24 ten thousands=24*10000=240,000
Generally the standard form of 24 ten thousands is
240,000
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Which statement is true?
This figure shows the procedure for constructing a
A. Pair of parallel lines
B. Bisector of an angle
C. Perpendicular from a point on a line
D. Perpendicular bisector
Answer:
B
Step-by-step explanation:
Prove that there exists a unique real number solution to the equation x3 + x2 − 1 = 0 between x = 2/3 and x=1
To prove that there exists a unique real number solution to the equation x³ + x² - 1 = 0 between x = 2/3 and x = 1, we can use the Intermediate Value Theorem and the property of the derivative.
Explanation:To prove that there exists a unique real number solution to the equation x³ + x² - 1 = 0 between x = 2/3 and x = 1, we can use the Intermediate Value Theorem.
First, we substitute x = 2/3 into the equation and get a negative value (-1/27). Then, we substitute x = 1 into the equation and get a positive value (1).
Since the function is continuous between x = 2/3 and x = 1, and it changes sign, there must exist a solution somewhere between them.
To show that the solution is unique, we can use the fact that the derivative of the function, 3x² + 2x, is always positive for all real numbers.
This implies that the function is strictly increasing, so it can only intersect the x-axis at one point.
One of the largest pumpkins ever grown weighed about 3/4 ton how many lb did the pumpkin weigh
Evaluate. 3−1⋅(4⋅6)⋅2−3 Enter your answer in the box.
The answer to this question is 1
The measure of an angle is 8 greater than 3 times its supplement. Find he measurement of the angle.
A tortoise is walking in the desert. It walks for 3 minutes at a speed of 7.5 meters per minute. For how many meters does it walk?
Use synthetic division and the Remainder Theorem to find P(a).
P(x) = x3 + 4x2 − 8x − 6; a = −2
−2
0
18
20
Answer:
c: 18
Step-by-step explanation:
The length of a rectangle is twice its width. if the area of the rectangle is 200 yd2 , find its perimeter.
The perimeter of the rectangle is 60 yards
What is the Perimeter of a Rectangle?
The perimeter P of a rectangle is given by the formula, P=2 ( L + W) , where L is the length and W is the width of the rectangle.
Perimeter P = 2 ( Length + Width )
Given data ,
Let the length of the rectangle be = a
Let the width of the rectangle be = b
The length of the rectangle is twice the width
So ,
a = 2b
The area of the rectangle is A = 200 yards²
The area of the rectangle is given by
Area of Rectangle = Length x Width
And , substituting the values for length and width , we get
Area of Rectangle = 2b x b
2b² = 200
Divide by 2 on both sides , we get
b² = 100
Taking square root on both sides , we get
b = 10
a = 2b
a = 20
Therefore , the length of the rectangle is 20 yards and width of the rectangle is 10 yards
Now , the perimeter of the rectangle is P = 2 ( length + width )
Perimeter of the rectangle P = 2 ( 10 + 20 )
= 2 x 30
= 60 yards
Hence , the perimeter of the rectangle is 60 yards
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A teacher has 100 pencils in a cup. 16 are more sharp than dull. How many are dull and how many are sharp?
what is the value of x in the equation 6= x/4 - 4
Answer:
c
Step-by-step explanation:
John wants to buy a watermelon that weighs 5.7 pounds.The watermelon is priced at $1.58 per pound.How much is the total cost of the watermelon
Select the postulate that is illustrated for the real numbers. 5 + (-5) = 0
The commutative postulate for multiplication
Multiplication by one
The addition inverse postulate
Commutative postulate for addition
The distributive postulate
The addition of zero postulate
The multiplication inverse
the answer is F.The addition of zero postulate
The addition inverse postulate explains the relationship between numbers like 5 and -5 that add up to zero, crucial for understanding real number properties.
The addition inverse postulate is illustrated by the expression 5 + (-5) = 0 for real numbers. This postulate states that for every number, there exists an additive inverse that when added together results in zero.
For example, in the given expression, 5 and -5 are additive inverses of each other, and when added, they cancel out to give zero, satisfying the addition inverse postulate.
This concept plays a crucial role in understanding the properties and operations of real numbers.
Perform the indicated computation. write the answer in scientific notation. 12x10^4
Which expression is equivalent to sin^2 x-sin x - 2/sin x -2
A. sin x + 1
B. sin x − 1
C. sin2x
D. -sin2x
harold serves himself 1 1/2 ounces serving of cereal each morning. How many servings does he get from a box of his favorite cereal?
What metamorphic rock has nonfoliated texture?
Answer:
Marble, quartzite, and soapstone.Step-by-step explanation:
Non-foliated metamorphic rocks are those which don't have a rough texture, like foliated ones, their composition gives them this characteristics: not having layers or banded appearance.
Last weekend Jane studied 4 2 3 hours for her history final and 4 1 4 hours for her math exam. How many hours in all did Jane study for the two tests?
Jane studied a total of 5 11/12 hours for her history final and math exam.
Explanation:To find the total number of hours Jane studied for the two tests, we add together the number of hours she studied for each test. Jane studied 4 2/3 hours for her history final and 4 1/4 hours for her math exam.
To add these fractions, we need to have a common denominator. The least common multiple of 3 and 4 is 12. We can convert 2/3 to 8/12 and 1/4 to 3/12.
Now we can add the fractions: 4 8/12 + 4 3/12 = 5 11/12.
Therefore, Jane studied a total of 5 11/12 hours for the two tests.
Final answer:
Jane studied a total of 7 3/4 hours for her history and math exams.
Explanation:
To find out how many hours Jane studied for the two tests, we simply need to add the hours she studied for each test together.
Jane studied 4 2/3 hours for her history final and 4 1/4 hours for her math exam.
We can add these two fractions by finding a common denominator and then adding the numerators.
In this case, the common denominator is 12. So, we have (4 * 4 + 2) / 3 = 18/3 and (4 * 3 + 1) / 4 = 13/4.
Adding these fractions together, we get (18/3) + (13/4) = 54/12 + 39/12 = 93/12. Simplifying this fraction, we get 7 3/4 hours.
Therefore, Jane studied a total of 7 3/4 hours for the two tests.