writing exponential functions (4, 112/81), (-1, 21/2)
Answers
The given points are (4, 112/81) and (-1, 21/2).
To find an exponential function from the given points, we have to use the forms.
[tex]\begin{gathered} y_1=ab^{x_1} \\ y_2=ab^{x_2} \end{gathered}[/tex]Now, we replace each point in each equation.
[tex]\begin{gathered} \frac{112}{81}=ab^4 \\ \frac{21}{8}=ab^{-1} \end{gathered}[/tex]We solve this system of equations.
Let's isolate a in the second equation.
[tex]\begin{gathered} \frac{21}{8}=\frac{a}{b} \\ \frac{21b}{8}=a \end{gathered}[/tex]Then, we replace it in the first equation
[tex]\frac{112}{81}=(\frac{21b}{8})\cdot b^4[/tex]We solve for b.
[tex]\begin{gathered} \frac{112\cdot8}{81\cdot21}=b\cdot b^4 \\ \frac{896}{1701}=b^5 \\ b=\sqrt[5]{\frac{896}{1701}}=\frac{2\sqrt[5]{4}}{3} \\ b\approx0.88 \end{gathered}[/tex]Once we have the base of the exponential function, we look for the coefficient a.
[tex]a=\frac{21b}{8}=\frac{21}{8}(\frac{2\sqrt[5]{4}}{3})=\frac{7\sqrt[5]{4}}{4}[/tex]Therefore, the exponential function is[tex]y=\frac{7\sqrt[5]{4}}{4}\cdot(\frac{2\sqrt[5]{4}}{3})^x[/tex]The image below shows the graph of this function.
Related Questions
Find the sum of the first three terms of the geometric series represented by the formula an = (825)(52)(n - 1).
Answers
ANSWER:
2nd option: 78/25
STEP-BY-STEP EXPLANATION:
We have the following geometric series:
[tex]a_n=\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^{\left(n-1\right)}[/tex]We calculate the sum, replace n by 1,2,3, just like this:
[tex]\begin{gathered} s_n=\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^{\left(1-1\right)}+\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^{\left(2-1\right)}+\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^{\left(3-1\right)} \\ s_n=\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^0+\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^1+\left(\frac{8}{25}\right)\cdot\left(\frac{5}{2}\right)^2 \\ s_n=\frac{8}{25}+\frac{4}{5}+\frac{8}{4} \\ s_n=\frac{32+80+200}{100} \\ s_n=\frac{312}{100} \\ s_n=\frac{78}{25} \end{gathered}[/tex]The sum of the first 3 terms is 78/25
which expressions are equivalent to 5(–2k – 3) + 2k?a) (-5•3)-8kb) -15c) none of em
Answers
Question 3: 12 ptsA circular pool is surrounded by a circular walkway. The radius of the pool is y - 4 and the radius of the full circleformed by the walkway is y + 4. Write a polynomial that represents the area of just the walkway itself, notincluding the space covered by the pool.The area of a circle is given by A = r7?, where r represents the radius of the circle.)O 16ny + 32O 16TyO-16nyO 32
Answers
Explanation
Step 1
the area of a circle is given by:
[tex]\begin{gathered} \text{Area}_c=\pi r^2 \\ \text{where r is the radius} \end{gathered}[/tex]so, the area of teh walkway will be the difference of areas
[tex]\begin{gathered} A_{walkway}=A_{entire\text{ circle}}-Area_{pool} \\ \text{replace} \\ A_{walkway}=\pi(y+4)^2-\pi(y-4)^2 \end{gathered}[/tex]Step 2
expand the polynomius:
[tex]\begin{gathered} A_{walkway}=\pi(y+4)^2-\pi(y-4)^2 \\ A_{walkway}=\pi(y^2+8y+16)^{}-\pi(y^2-8y+16) \\ A_{walkway}=\pi(y^2+8y+16)^{}-\pi(y^2-8y+16) \\ A_{walkway}=\pi(y^2+8y+16-(y^2-8y+16)) \\ A_{walkway}=\pi(y^2+8y+16-y^2+8y-16)) \\ A_{walkway}=\pi(16y) \\ \end{gathered}[/tex]therefore, the answer is
[tex]16\text{ }\pi\text{ y}[/tex]I hope this helps you
The average price of a new home in a neighborhood in thousands of dollars (f(x)) is related to the number of years since the neighborhood was built (x) in the function: Use the graph of the function to describe its domain.A. The domain is the average price of homes in a new neighborhood, which is represented by all real numbers from 0 to 100B. The domain is the average price of homes in a new neighborhood, which is represented by all real numbers from 0 to infinity C. The domain is the number of years since the neighborhood was built, which is represented by all real numbers from zero to infinityD. The domain is the number of years since the neighborhood was built, which is represented by all real numbers from 0 to 100l
Answers
As given by the question
There are given that the graph of the function.
Now,
According to the domain concept, the domain is defined for the input valuewhich is given in the x-axis.
That means, all x-axis input value range is called domain.
Then,
From the given graph:
The domain is the number of years since the neighborhood was built, that represent the range of x-axis 0 ti infinity.
Hence, the correct option is C.
While playing golf, Maurice hits the golf ball and it travels 361.87 feet. Assume the golf ball travels the same distance everyTime they hit it. Estimate the total amount of distance the ball will travel after 15 hits.Round the distance traveled each time the golf ball was hit to the nearest ten feet before calculating
Answers
Answer:
5400 feet
Explanation:
The distance the ball travels each time it was hit = 361.87 feet
First, this distance is rounded to the nearest ten feet.
[tex]361.87\approx360\:feet[/tex]Multiply 360 by 15 hits:
[tex]360\times15=5400\:feet[/tex]The total amount of distance the ball will travel after 15 hits is 5400 feet.
$72 for 7/1/2 hours Part A Find the unit rate. How much would it be for 40 hours?
Answers
We have that for seven and a half hours, you get 72$, then we have the following rule of three:
[tex]\begin{gathered} 72\rightarrow7\frac{1}{2}hours \\ x\rightarrow1hour \end{gathered}[/tex]then we have that:
[tex]\begin{gathered} x=\frac{72\cdot1}{7\frac{1}{2}}=\frac{72}{\frac{15}{2}}=\frac{144}{15}=9.6 \\ x=9.6 \end{gathered}[/tex]therefore, for each hour, you get $9.6
So, for 40 hours we have:
[tex]40\cdot9.6=384[/tex]finally, we have that for 40 hours you get $384
Multiply. -8. -9/3 . 2/-5Write your answer in simplest form.
Answers
To multiply these numbers, the first step is to writhe "-8" as an improper fraction, to do so, divide it by 1
[tex](-\frac{8}{1})\cdot(-\frac{9}{3})\cdot(-\frac{2}{5})[/tex]Next is to solve the multiplication, to do so, first multiply the first two terms of the multiplication:
[tex](-\frac{8}{1})\cdot(-\frac{9}{3})[/tex]The multiplication is between two negative numbers, when you multiply two negative numbers, the minus signs cancel each other and turn into a positive value, this is called "double-negative"
[tex](-\frac{8}{1})\cdot(-\frac{9}{3})=\frac{8\cdot9}{1\cdot3}=\frac{72}{3}[/tex]Next multiply the result by the third fraction -2/5
This time you are multiplying a positive and a negative number, so the result of the calculation will be negative
[tex]\frac{72}{3}\cdot(-\frac{2}{5})=-\frac{72\cdot2}{3\cdot5}=-\frac{144}{15}[/tex]Final step is to simplify the result, both 144 and 15 are divisible by 3, so divide the numerator and denominator by 3 to simplify the result to the simplest form:
[tex]-\frac{144\div3}{15\div3}=-\frac{48}{5}[/tex]Please see the photo below. Please draw the photo on a piece of paper or computer/laptop. Thank you.
Answers
The Solution:
Given:
Required:
To construct a bisector of each of the given lines.
Steps:
1. Take your compass and put the pin on one end of the line, and then expand the compass to at least more than half the length of the line ( but not greater than the length of the line).
2. Make an arc on the upper side and the lower side of the middle of the line, and then repeat the process when you take the pin mouth of the compass to the other end of the given line.
3. Connect the pairs of intersections of the arcs to make a straight line.
The straight is the required bisector.
Below is an example with the first line:
Write an equation for a line going through the point (-5, -10) that is parallel to theline 1/5x-1/6y = 7.
Answers
Two lines are parallel if they have the same slope. In order to better visualize the slope of the line we will express it in the slope-intercept form, which is done below:
[tex]\begin{gathered} \frac{1}{5}x-\frac{1}{6}y\text{ = 7} \\ \frac{1}{5}x-7=\frac{1}{6}y \\ \frac{1}{6}y\text{ = }\frac{1}{5}x-7 \\ y\text{ = }\frac{6}{5}x\text{ - 42} \end{gathered}[/tex]We now know that the slope of the line is 6/5, because in this form the slope is always the number that is multiplying the "x" variable. So we need to find a line of the type:
[tex]h(x)\text{ = }\frac{6}{5}x+b[/tex]Therefore the only needed variable is "b", which we can find by applying the known point (-5, -10).
[tex]\begin{gathered} -10\text{ = }\frac{6}{5}\cdot(-5)\text{ + b} \\ -10=-6+b \\ b=-10+6 \\ b=-4 \end{gathered}[/tex]The expression of the line is then:
[tex]h(x)\text{ = }\frac{6}{5}x-4[/tex]Which of the following best represents the graph of a line with an undefined slope?
Answers
we know that
The slope is undefined, when we have a vertical line
therefore
the answer is the option 4 (vertical line)Find the distance between the two points in simplest radical form.(8,6) and (3,−6)
Answers
Given
Two points (8,6) and (3,−6)
Find
distance between the two points
Explanation
Distance between the two points is given by
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]so , distance between (8,6) and (3,−6) is
[tex]\begin{gathered} d=\sqrt{(3-8)^2+(-6-6)^2} \\ d=\sqrt{25+144} \\ d=\sqrt{169} \\ d=13 \end{gathered}[/tex]Final Answer
Therefore , the distance between these two points is 13
Determine if the following equations are parallel, perpendicular, or neither. 5(x + 3) = 3y + 12 and 5x + 3y = 15
Answers
Solution
We have the following equation:
5(x+3)= 3y+12 (1)
Solving for y we got:
3y= -12+ 5(x+3)
3y = -12 + 5x+15
3y= 5x +3
y= 5/3 x +1
The slope for the first case is: m1= 5/3
5x + 3y = 15 (2)
Solving for y we got:
3y= 15-5x
y= 5 -5/3x
The slope is given by : m2= -5/3
Then m1*m2 is not equal to -1 (NOT perpendicular)
m1 is different from m2 (NOT parallel)
Then are not perpendicular or parallel
Lionfish are considered an invasive species, with an annual growth rate of 69%. A scientist estimates there are 9,000 lionfish in a certain bay after the first year.
Answers
Given:
Lionfish are considered an invasive species, with an annual growth rate of 69%. A scientist estimates there are 9,000 lionfish in a certain bay after the first year.
The general equation of the growth is:
[tex]P(t)=P_0\cdot(1+r)^t[/tex]Given rate = r = 69% = 0.69
After 1 year, P = 9000
Substitute to find the initial number of Lionfish
So,
[tex]\begin{gathered} 9000=P_0\cdot(1+0.69)^1 \\ 9000=P_0\cdot1.69 \\ P_0=\frac{9000}{1.69}\approx5325 \end{gathered}[/tex]Part (A), we will write an explicit formula f(n) that represents the number of lionfish after n years
so, the formula will be:
[tex]f(n)=5325\cdot1.69^n[/tex]Part (B): we will find the number of lionfish after 6 years
so, substitute with n = 6 into the equation of part (a)
[tex]f(6)=5325\cdot1.69^6=124,073[/tex]So, after 6 years, the number of lionfish = 124,073
Part (C): The scientists remove 1400 fish per year after the first year
So, we the number of lionfish:
[tex]9000-1400=7600[/tex]Then after 2 years, the number of lionfish
[tex]7600\cdot1.69-1400[/tex]After 3 years:
[tex]\begin{gathered} (7600\cdot1.69-1400)\cdot1.69-1400 \\ =7600\cdot1.69^2-1400\cdot1.69-1400 \\ =7600\cdot1.69^2-1400\cdot(1+1.69) \end{gathered}[/tex]So, after (n) years:
[tex]7600\cdot1.69^{n-1}-1400\cdot(1+1.69)^{n-2}^{}[/tex]The worktop is to be covered with square tiles each measuring 4cm by 4cm. How many tiles are needed to cover the worktop.
PLease help and show clear explaination and answer now
ITS FOR LEVEL 2 GET DIFFERENT ANSWERS
Answers
1900 tiles are needed to cover the worktop.
From the question, we have
The surface area of the worktop is 3.04m squared.
The worktop is to be covered with square tiles each measuring 4cm by 4cm.
Number of tiles = (3.04*10000)/(4*4)
=1900
Multiplication:
To determine the sum of two or more numbers, mathematicians multiply the numbers. It is a basic mathematical procedure that is widely used in daily life. Multiplication is used when we need to mix groups of like sizes. Multiplication is a representation of the underlying idea of adding the same number repeatedly. The outcome of multiplying two or more numbers is referred to as the product of those numbers, and the factors are the quantities that are multiplied. Multiplying the numbers makes it simpler to add the same number repeatedly.
Complete question:
The worktop is to be covered with square tiles, each measuring 4 cm by 4 cm. How many tiles are needed to cover the worktop?The surface area of the worktop is 3.04m squared.
To learn more about multiplication visit: https://brainly.com/question/5992872
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write the expanded form of the expression : 7(2x + y)
Answers
ANSWER
14x + 7y
EXPLANATION
We want to write the expanded form of the expression given:
7(2x + y)
To do this, we have to use the distribution property by using the number outside the bracket to multiply each of the terms in the bracket.
So, we have that:
7(2x + y) = (7 * 2x) + (7 * y)
= 14x + 7y
That is the answer.
A bag contains 6 green balls and 4 yellow balls. What is the probability that two balls picked randomly are both of the same color?
Answers
Given:
The number of green balls is G = 6.
The numer of yellow balls is Y = 4.
Explanation:
Determine the total number of balls.
[tex]\begin{gathered} T=6+4 \\ =10 \end{gathered}[/tex]Determine the probability for both selected balls to be green.
[tex]\begin{gathered} P(G)=\frac{6}{10}\cdot\frac{5}{9} \\ =\frac{15}{45} \end{gathered}[/tex]Determine the probability for selected balls to be yellow.
[tex]\begin{gathered} P(Y)=\frac{4}{10}\cdot\frac{3}{9} \\ =\frac{6}{45} \end{gathered}[/tex]Determine the probability for both selected balls to be of same colour.
[tex]\begin{gathered} P=P(G)+P(Y) \\ =\frac{15}{45}+\frac{6}{45} \\ =\frac{21}{45} \\ =\frac{7}{15} \end{gathered}[/tex]4 i Rotate the figure 90° counterclockwise about the origin, and then reflect in the x-axis. Polygon 1. Move PREV 1 2 3
Answers
to rotate 90 degrees counterclockwise we must transform the points like this
[tex](x,y)\longrightarrow(y,-x)[/tex]and then invert the sign of the y-coordinate or the second coordinate of each point, so the total transformation is
[tex](x,y)\longrightarrow(y,x)[/tex]now, transform each point
[tex](0,4)\longrightarrow(4,0)[/tex][tex](0,1)\longrightarrow(1,0)[/tex][tex](2,1)\longrightarrow(1,2)[/tex][tex](2,4)\longrightarrow(4,2)[/tex]order the numbers -7,7,1 and -1 from least to greatest.
Answers
as we move towards the right,the value of the numbers on a number line increases. The numbers to the left of zero are negative while the numbers to the right of zero are positive.
Therefore, by ordering the numbers from least to greatest, it would be
- 7, - 1, 1, 7
A 14 foot ladder is leaning against a building. The ladder makes a 45 degree angle with the building. How far up the building does the ladder reach?A. 14,2 feetB. 7 feetc. 28/2 feetD. 7,2 feet
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Answer: The problem can be visualized with the help of the following diagram:
Therefore the building height can be determined by using the pythagorean theorem, the steps are as follows:
[tex]\begin{gathered} x^2+x^2=14^2 \\ \\ 2x^2=14^2 \\ \\ x=\sqrt{\frac{14^2}{2}}=\sqrt{98} \\ \\ h=x=9.899ft \\ \\ \end{gathered}[/tex]Therefore the ladder reaches 9.9ft up the wall.
Which graph best represent a line perpendicular to the line of the equation y= -1/3x - 7 ?
Answers
The equation of the given line is
[tex]y=-\frac{1}{3}x-7[/tex]Where: The slope is -1/3
Perpendicular lines have additive reciprocal slopes which means if the slope of one of them is m, then the slope of the other is -1/m
Then the slope of the perpendicular line to the given line is 3
So, we have to look for the graph of positive slope
The graphs of A and D have positive slopes because the directions of the lines are increasing from left to right
Then we have to find the slope of each line to find the correct choice
Since the slope of the line is 3, then the y part increases 3 units for 1 part increases of x
We can see that in graph A
The answer is A
What is the difference between the decimal forms of rational numbers and the decimal forms of irrational numbers?
Answers
The decimal forms of rational numbers can be finite, i.e. have a non infinite number of digits. In case they have infinite digits then these are periodic. This means that there's a patron of digits that is repeated infinitely.
Irrational numbers on the other hand can only have infinite digits on their decimal form and they are not periodic.
x+2x+5=x+19please help
Answers
To solve this equation
Step 1:
x + 2x + 5 = + 19
Lisa receives a net pay of $619.06 biweekly. She has $143withheld from her pay each pay period. What is her annual gross salary?a. $ 762.06b. $18,289.44c. $19,813.56d. $39,627.12
Answers
Answer:
c. $19,813.56
Explanation:
Given:
• Lisa receives a net pay of $619.06 biweekly.
,• $143 is withheld from her pay each pay period.
We are required to find her annual gross salary.
First, determine her gross salary for each pay period.
[tex]\begin{gathered} \text{Gross Salary}=\text{Net Pay+Deduction} \\ =619.06+143 \\ =\$762.06 \end{gathered}[/tex]Next, determine the number of payment periods.
[tex]\begin{gathered} \text{Lisa is paid biwe}ekly,\text{ that is every 2 weeks.} \\ The\text{ number of weeks in a year}=52 \\ \text{Therefore:} \\ \text{The number of payment periods}=\frac{52}{2}=26 \end{gathered}[/tex]Finally, multiply her gross salary per period by the number of periods to get her annual gross salary.
[tex]\begin{gathered} \text{Gross annual salary}=26\times762.06 \\ =\$$19,813.56$ \end{gathered}[/tex]Lisa's annual gross salary is $19,813.56.
Option C is correct.
The graph is shifted 1 unit down and 4 units left
Answers
To answer this question, we need to remember the rules of transformation of functions, these rules are shown below:
Using these rules, we have that the equation that represents the new graph is:
[tex]y=\sqrt[3]{x+4}-1[/tex](−2,1) is a solution to the following system of linear equations6−3=−152+=−3
Answers
We have the following system of linear equations:
[tex]\begin{gathered} 6x-3y=-15 \\ 2x+y=-3 \end{gathered}[/tex]We want to know if the pair (x,y) = (-2,1) is a solution of the system above.
To see if this pair is a solution, we simply replace the values of x and y in the equations above and we verify if the equality holds.
1) Replacing in the first equation we see that:
6x - 3y = 6*(-2) - 3*(1) = -12 -3 = -15 ✓
The equality holds.
2) Replacing in the second equation we see that:
2x + y = 2*(-2) + 1 = -4 + 1 = -3 ✓
The equality holds.
We conclude that (-2,1) is a solution to the system of linear equations.
Answer: True
A graph shows three linear relationships but different y y-intercepts the following slopes line1: 1 / 5 line 2: 3/5 line 3: 6 / 5 write an equation for each line type your answers in the boxes below
Answers
Explanation
Step 1
we have 3 lines, the slopes and the Y-intercept( or a point of the line)
use :
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ \text{where} \\ P1(x_1,y_1) \\ \text{and m is the slope} \end{gathered}[/tex]Step 2
Let
[tex]\begin{gathered} \text{slope}=\frac{1}{5} \\ P(0,5)\text{ gr}een \\ \text{replacing} \\ y-5=\frac{1}{5}(x-0) \\ y=\frac{1}{5}x+5\text{ Equation(1) ( gre}en) \end{gathered}[/tex]Step 3
[tex]\begin{gathered} \text{slope}=\frac{3}{5} \\ P(0,7) \\ \text{replacing} \\ y-y_1=m(x-x_1) \\ y-7=\frac{3}{5}(x-0) \\ y=\frac{3}{5}x+7\text{ function (2) Blue} \end{gathered}[/tex]Step 4
[tex]\begin{gathered} \text{slope}=\frac{6}{5} \\ P(0,3)\text{red} \\ \text{replacing} \\ y-y_1=m(x-x_1) \\ y-3=\frac{6}{5}(x-0) \\ y=\frac{6}{5}x+3\text{ Function (3) red} \end{gathered}[/tex]I hope this helps you
Without dividing how can you decide whether the quotient of 7.16 ÷ 4 will be less than or greater than 2
Answers
Answer:
Hope this helps : )
Step-by-step explanation:
We know that the quotient of 7.16 ÷ 4 can be multipled with 4 to get 7.16. So if we multiply 4 × 2, then the product is 8. Now we know that the quotient of 7.16 ÷ 4 is less than 2.
Check:
7.16 ÷ 4 = 1.79
1.79 < 2 ✓
D (x - 4) B . In the figure shown, what is the value of x? C •E 19 A F
Answers
x = 23
Explanation:Given: triangle ABC and triangle DEF
we need to find the triangle congruency theorem in order to determine the value of x.
AB = DE
AC = DF
∠A = ∠D
the sides BC and EF respectively were not marked.
Since, we were not given the value of the angles but we know they are equal. And two sides of triangle ABC and the corresponding two sides of triangle DEF were given. It means BC is equal to EF.
The sides opposite ∠A = BC
The sides opposite ∠D = EF
BC = EF
x - 4 = 19
collect like terms:
x = 19 + 4
x = 23
A glider files 8 miles south from the airport and then 15 miles east. Then it files in a straight line back to the airport. What was the distance of the glider's last leg back to the airport ?
Answers
The schematic diagram below represents the path followed by the glider,
The point A represents the location of the airport.
Observe that the path of the glider forms a right angled triangle ABC.
So the hypotenuse AC can be calculated by using Pythagoras Theorem as,
[tex]\begin{gathered} AC^2=AB^2+BC^2 \\ AC^2=(8)^2+(15)^2 \\ AC^2=64+225 \\ AC^2=289 \\ AC^2=17^2 \\ AC=17 \end{gathered}[/tex]Thus, the distance of the glider's last leg back to the airport is 17 miles.
So the second option is the correct choice.
Convert each equation to slope-intercept form. Then label the slope & y-intercept.
Answers
C. The equation is
[tex]4x-6y=18[/tex]An equation is in slope-intercept form if it is in the form
[tex]y=mx+c[/tex]Expressing the given equation in slope-intercept
This gives
[tex]\begin{gathered} 4x-6y=18 \\ -6y=-4x+18 \end{gathered}[/tex]Divide through by -6
This gives
[tex]\begin{gathered} -\frac{6y}{-6}=-\frac{4x}{-6}+\frac{18}{-6} \\ y=\frac{2}{3}x-3 \end{gathered}[/tex]Therefore, the slope-intercept form of the given equation is
[tex]y=\frac{2}{3}x-3[/tex]
Where
slope = 2/3
y-intercept = -3