Let the dimensions of the pen be L for the length and W for the width. The area of the pen is L*W.
We are given that the width of the pen is 80 meters and that the length of the pen is greater than the width. Therefore, the length of the pen must be greater than 80 meters.
Since the area of the pen is L*W, the area will be greatest when L is as large as possible and W is as large as possible. Therefore, we can set L equal to 80 and W equal to 80 to find the dimensions of the pen that give the greatest area:
L = 80
W = 80
The dimensions of the pen that give the greatest area are 80 meters by 80 meters. Therefore, the answer is 80m x 80m.
The dimensions of the pen are 20 m × 40 m and the area will be 600 m².
What is the area?An object's area is how much space it takes up in two dimensions. It is the measurement of the number of unit squares that completely encompass the surface of a compact figure. The squared unit, which is frequently expressed as inches, square feet, etc., is the accepted unit of area.
As per the given data in the question,
Let the length of the rectangle is x and,
Let the width of the rectangle is y.
So, the equation will be,
2x + y = 80 (1)
y = 80 - 2x (2)
Use the formula of the area of a rectangle,
A = x × y
Substitute y = 80 - 2x in the above formula,
A = x × (80 - 2x)
= 80x - 2(x)²
= -2x² + 80x
dA/dx = -2x² + 80x
A' = -4x + 80
-4x = -80
x = 80/4
x = 20 m
Substitute x = 20 in equation (2)
y = 80 - 2(20)
y = 40 m
So, the area should be,
A = 20 × 40
A = 600 m².
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30-60-90 triangle with a measure of 10 (longer leg) find the hypotenuse and shorter leg
For given 30-60-90 triangle, the measure of the shorter leg = 5.78 units and hypotenuse = 11.55 units
In this question we have been given 30-60-90 triangle with a measure of 10 (longer leg)
We need to find the hypotenuse and shorter leg.
We know that longer leg of 30-60-90 triangle is opposite to angle 60
Consider the sine of angle 60
sin(60) = 10/hypotenuse
(√3)/2 = 10/hypotenuse
hypotenuse = 10 * (2/√3)
hypotenuse = 11.55 units
By Pythagoras theorem,
hypotenuse² = (longer leg)² + (shorter leg)²
11.55² = 10² + (shorter leg)²
(shorter leg)² = 11.55² - 10²
shorter leg = √(33.4025)
shorter leg = 5.78 units
Therefore, shorter leg = 5.78 units and hypotenuse = 11.55 units
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please help‼️
Write the vertex form for the quadratic function f, whose vertex is (4, 9) and has leading coefficient
a=2.
Answer:
y=2(x-4)^2+9
Step-by-step explanation:
Vertex form: y=a(x-h)^2+k
The vertex represents the h and k variables.
Bill is in a hot air balloon that has just taken off and is floating above its launching point.
Lillian is standing on the ground, 9 meters away from the launching point. If Bill and Lillian
are 15 meters apart, how high up is Bill?
meters
Submit
Two-fifths of the students in the music class of twenty students were boys. How many boys are in the music class?
Answer:
8
Step-by-step explanation:
the answer is 8
Answer:
8
Step-by-step explanation:
total students=20
boys=2/5×20
=8
A newspaper vendor sold 2000 newspapers each at sh.40.He was paid sh.2 per each newspaper as commission.What was the percentage commission.
.
Answer:
5 percent
Step-by-step explanation:
100/40=2.5
2.5*2=5
If tan(x) = 0.4, find tan(x+kπ), where k∈Z
a. -0.4
b. -0.4k
c. 0
d. 0.4
e. answer not possible
(please answer with explanation)(80 pts)
tan(x + kπ) = 0.4 where k∈Z
What do you mean by trignometry?
Trigonometry is one of the important branches in the history of mathematics that studies the relationship between the sides and angles of right triangles.
Trigonometric ratios of triangles are also called trigonometric functions. Sine, cosine, and tangent are three important trigonometric functions, abbreviated sin, cos, and tan
It is given that tan(x) = 0.4
Also, tan(x + kπ) = tanx where k∈Z
Therefore, tan(x + kπ) = 0.4 where k∈Z
Hence, option D is correct.
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Use the binomial formula to find the coefficient of the y^21p^3 term in the expansion of (y+3p)^24
Answer: normal formula define the cup full of is Q -2+ you and me 32,000 you’re welcome
Step-by-step explanation:
what are the solutions of 2x^2 + 3x = -5
Answer:
No solution
Step-by-step explanation:
The square root of a negative number is not a real number
Last week, Vernon worked the following days and times:
Day 1: 5 hours, 15 minutes
Day 2: 6 hours, 30 minutes
Day 3: 3 hours, 45 minutes
Day 4: 4 hours, 0 minutes
Find Vernon's total time worked in decimal hours.
20.00
18.90
18.45
19.50
The total time worked by Vernon in decimal hours is 19.50 hours.
How to find the total time worked in decimal?Last week, Vernon worked the following days and times:
Day 1: 5 hours, 15 minutesDay 2: 6 hours, 30 minutesDay 3: 3 hours, 45 minutesDay 4: 4 hours, 0 minutesTherefore, the total time Vernon worked in decimal hours can be calculated as follows:
We have to sum the whole time Vernon worked in the 4 days.
Therefore, we have to convert all the minutes to hours.
total time worked = 5.25 + 6.5 + 3.75 + 4
total time worked = 19.50 hours
Therefore,
total time worked by Vernon = 19.50 hours
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1. A taxi charges $2.00 for up to and including the first mile and $1.60 for each mile thereafter. Which equation best fits this situation, where x
represents the miles after the first mile and y is the total amount.
A)y=1.602 - 2.00
B)y=2.002 -1.6
C)y =2.00x + 1.60
D)y=1.60x+2.00
Answer:
D
Step-by-step explanation:
The taxi charges $2.00 for any distance less than or up to 1 mile.
Then for every mile after the taxi will charge you a fee of $1.60 dependent upon how many miles you go.
This can be modeled as:
y = 2.00 +1.60(x)
where x is the number of miles after the first mile.
help meeeeeeeeeee pleaseee
Answer: 51.85
Step-by-step explanation: you use the formula given and plug is 25 for t because it is the number of years. So you get 200*e^(-0.054*25) = 51.85
Please see attached problem
At December 31, the records of Nortech Corporation provided the following selected and incomplete data:
Common stock (par $1; no changes during the current year).
Shares authorized, 500,000.
Shares issued, 2; issue price $12 per share.
Common Stock account, $100,000.
Shares held as treasury stock, 2,000 shares, cost $10 per share.
Net income for the current year, $24,500.
Dividends declared and paid during the current year, $14,700.
Retained Earnings balance, beginning of the year, $145,000.
Required:
Complete the following: TIP: To determine the number of shares issued, divide the balance in the Common Stock account by the par
value per share. (Round "per share" answers to 2 decimal places.)
1-a. Shares authorized
1-b. Shares issued
1-c. Shares outstanding
2. The balance in Additional Paid-in Capital would be
3. Earnings per share is
4. Dividends paid per share of common stock is
5. Treasury stock should be reported in the stockholders' equity section of the balance
sheet in the amount of
6. Assume that the board of directors approved a 2-for-1 stock split. After the stock split,
the par value per share will be
500,000
290,000
The solutions for the given questions are given below:
What is meant by income?Earnings from employment, investments, or company operations. Sam, for instance, had a weekly income of $770 after receiving $30 in bank interest, $700 from his employment, and an additional $40 from buying and selling veggies.
Wages is the amount of money obtained through employment; income is the entire amount of money obtained, which includes earnings, benefits, pensions, and other types of payments.
1.a 500000 shares have been authorized.
1/(210000) = 210000 shares issued Number of shares issued [Common stock account balance / par value of each share]
1-c) The number of shares outstanding is 210000 - 3100, or 206900 shares.
2) The APC Account Balance is $4620000, which is calculated as 210000 * (23 - 1).
3) Net income divided by the number of outstanding shares equals 165520 divided by 206900, or 0.8 per share.
4) The dividend per common share was paid at 53794/206900, or 0.26 per share.
5) 3100 * 21 = $65,100 should be stated for Treasury Stock in the Stockholders' Equity part of the Balance Sheet.
6) Following the stock split, the par value of each share is equal to 1 / 2 ($0.50 per share).
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Sammy is playing a board game. He rolls two number cubes, each numbered 1-6. If he rolls a sum of 2 he wins $50; otherwise he loses $5. How much should Sammy expect to win or lose on average per roll?
he is supposed to lose an average of 1.36$ per roll
Answer:
Sammy loses $3.47 on average
Step-by-step explanation:
When you roll two number cubes, the number of possible combinations is 36 ( 6 * 6 ).
The only way to get a sum of two is rolling snake eyes (two ones), which means there's only one combination that gets this sum.
The chances of winning is 1/36 whilst the chance of losing is 35/36
$50*(1/36) - $5(35/36) = $1.39 - $4.86 = -$3.47
which means Sammy loses $3.47 on average
In the diagram, A is a point on the circumference of a circle with center O and radius r. A circular are with center A meets the circumference at B and C. The angle O A B is θ radians. The shaded region is bounded by the circumference of the circle and the are with center A joining B and C. The area of the shaded region is equal to half the area of the circle.
The area of the shaded region is equal to half the area of the circle.
What is the area of the circle?
The approximating parallelograms arbitrarily close and approximate the area of the circle by cutting it into ever-increasing slices.
The radius of the arc BC is |AB|=2rcosθ, and ∡AOB=π−2θ.
Dashed area= Area of sector BAC+ 2(Area of sector BOA−Area of triangleBOA)
= 1/2|AB|²(2θ)+2[1/2r²(π−2θ)−1/2r²sin(π−2θ)]
On the other hand, the area of the dashed surface is 1/2πr², then
1/2|AB|²(2θ)+2[1/2r²(π−2θ)−1/2r²sin(π−2θ)] = 1/2πr²
(2rcosθ)2(θ) + r²[π−2θ−sin(π−2θ)] = 1/2πr²
4θcos2θ+π−2θ−sin2θ = 1/2πr²
2θ(2cos2θ−1) = sin2θ−1/2π
2θcos2θ = (2sin2θ−π)/2
cos2θ = (2sin2θ−π)/4θ
Hence, the area of the shaded region is equal to half the area of the circle.
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Write the equation of the line in fully
simplified slope-intercept form.
12
11
10
96
8
7
6
-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
5
4
3
2
1
-2
2 3 4 5 6 78 0
y
-9
-10
-11
-12
1
2 3 4 5 6 7 8 9 10 11 12
X
Answer:
y=5/2x-4
Step-by-step explanation:
y=mx+b
5/2 is the slope
-4 is the y intercept
Please help - if could throw this out the window I would
The speed of the car will be 25.41 m/h
What is speed and velocity ?Velocity is the pace and direction of an object's movement, whereas speed is the time rate at which an object is travelling along a path. In other words, velocity is a vector quantity, whereas speed is a scalar quantity
According to the given information
d = [tex]\frac{s^{2} }{10}[/tex] + s
We are given that
distance is 90 feet
So, the expression becomes
90 = [tex]\frac{s^{2} }{10}[/tex] + s
[tex]s^{2}[/tex] + 10s - 900 = 0
which is quadratic in s
So applying the discriminant formula
[tex]$$[/tex]s [tex]=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}[/tex]
[tex]$$[/tex]and then we input the values:
s [tex]=\frac{-10 \pm \sqrt{(10)^2-4(1)(900)}}{2(1)}$$[/tex]
and solve:
s [tex]=\frac{-10 \pm \sqrt{3700}}{2}$$[/tex]
It will have two answers
s = -35.41 m/h
s = 25.41m/h
Speed cannot be negative
So
The speed of the car will be 25.41 m/h
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If you draw a random card from a standard deck of 52 cards, what are the odds of drawing the jack, queen, king, or ace of clubs? Write your answer in the form a:b as a fully reduced ratio. Note: In a standard deck of cards, each of the four suites has three face cards.
Answer:
12:52
Step-by-step explanation:
almost every card game has 52 cards, you have a 23% chance to draw a face card of any kind.
There are a king, queen, and jack for each of the four different suits: hearts, spades, clubs, and diamonds. The four suits have 13 cards each, for a total of 52 cards. Therefore, your chance of drawing a face card out of a deck of cards upon choosing the card at random is 12 out of 52, or about 23%.
The rectangular floor of a classroom is 34 feet in length and 22 feet in width. A scale drawing of the floor has a length of 17 inches. What is the perimeter, in inches, of the floor in the scale drawing?
The perimeter of the scale floor is 56 inches.
What is Perimeter?The perimeter of a shape is defined as the total distance around the shape. It is the length of the outline or boundary of any two-dimensional geometric shape.
To solve, we can set up fractions:
34/17 = 22/x
Cross multiply and you get:
34x = 374
Divide each side by 34:
x = 11
Since we are trying to find the perimeter of the floor, you can use this equation:
2w + 2l = p
Substitute to solve:
2 ( width = 11 )
2 ( length = 17 )
2 ( 17 ) + 2 ( 11 ) = p
34 + 22 = p
p = 56
Hence, the perimeter is 56inches
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determine 3x4 2x5 if x1, x2, x3, x4, and x5 satisfy the system of equations given below: 2x1 x2 x3 x4 x5
On solving the linear equations we get the result as
x1 + x2 − 2x3 + 3x4 + 2x5 = 6
3x1 + 3x2 − x3 + x4 + x5 = 12
2x1 + 2x2 − x3 + x4 − 2x5 = 5
4x1 + 4x2 + x3 +0x4 − 3x5 = 16
8x1 + 5x2 − 2x3 − x4 + 2x5 = 29
where ,
x1 = 5,x2 = -1, x3 = 3 ,x4 = 2 , x5 = 1
Finding the value of an equation's unknown variable can be done by solving the equation. If an equation contains a "equal to" sign, it is considered to be balanced. This equation so shows that the two quantities on both sides of it are equal. Left hand side and right hand side are the two sides of the equation (Right hand side).
One equation is x - 4 = 5, for instance. It illustrates that x - 4 (LHS) = 5. (RHS). Here, x is an unknowable quality or variable. Therefore, in order to determine the value of x, we must solve this equation.
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What is the value of i21
i
21
?
From the properties of a complex number, it was found that the value for [tex]i^{21}[/tex] is i.
Complex NumberA complex number is represented by the following form: a+bi, where a and b are real numbers. The variables: a is the real part and bi imaginary part. See an example: 6 + 13i , then: 6 represents the real part and 14i represents the imaginary part.
Regarding operations math, the same properties used for real numbers can be applied to complex numbers. And for solving the operations math with these numbers, it is important to know that i²= -1. Thus, 6i² = 6*(-1)= -6.
The question gives [tex]i^{21}[/tex], thus.
[tex]i^{21}=i^{20}*i[/tex]
[tex]i^{21}=(i^{2})^{10}*i\\ \\ i^{21}=(-1)^{10}*i\\ \\ i^{21}=1*i=i[/tex]
Therefore, the result for the question is i.
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Jerry builds 5 shelves complete the equations to show the area of shelf A
The area of the shelf A is 30 square feet
How to determine the area of the shelf AFrom the question, we have the following parameters that can be used in our computation:
Dimensions = 3 ft and 2 ft
The above parameters mean that
Length = 3 ft
Width = 2 ft
Number of shelves = 5
The area of the shelf A is then calculated as
Area = Number of shelves * Length * Width
Substitute the known values in the above equation, so, we have the following representation
Area = 5 * 3 * 2
Evaluate the products
So, we have
Area = 5 * 6
Evaluate the products
So, we have
Area = 30
Hence, the area is 30 square feet
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What is the name of the point located at
(
4
,
3
)
?
A coordinate plane has an x-axis from 0 to 5 in increments of 1 and a y-axis from 0 to 5 in increments of 1. Point M is plotted 3 units above the origin. Point T is plotted 5 units above the origin. Point Y is plotted 1 unit to the right of the origin. Point Z is plotted 1 unit to the right and 4 units above the origin. Point N is plotted 1 unit to the right and 5 units above the origin. Point L is plotted 2 units to the right and 2 units above the origin. Point W is plotted 3 units to the right and 2 units above the origin. Point P is plotted 4 units to the right of the origin. Point X is plotted 4 units to the right and 3 units above the origin. Point O is plotted 5 units to the right and 4 units above the origin. Point S is plotted 5 units to the right and 5 units above the origin.
A.
Point M
B.
Point P
C.
Point W
D.
Point X
The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.
The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.
The name of the point located at (4, 3) is X.
Option D is the correct answer.
What are coordinates in a graph?The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.
The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.
We have,
The coordinates of a point = (4, 3)
Point (4, 3) is located on the graph at point X.
(4, 3) means 4 is on the x-axis and 3 is on the y-axis.
The meeting point is point X.
Thus,
The name of the point located at (4, 3) is X.
Option D is the correct answer.
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18. Consider the following picture of a house.
Find:
(a) Its volume.
(b) Its surface area.
The volume of the picture is given as 24000 ft square
The surface area of the shape is 5360 ft square
How to solve for the volumeWe would have to solve for the volume of a cuboid + the volume of a triangular prism
We are aware that the prism's volume is equal to its base area divided by its length. As a result, the formula Volume of triangular prism = (1/2) bh L is used to get the volume of the prism in this instance.
Such that we would have: (1/2) bh L
where b = 30
h = 8
L = 50
when we put the values we would have
1 / 2 x 30 x 8 x 50
= 6000
The volume of a cuboid is l x b x h
= 30 x 50 x 12
= 18000
18000 + 6000 = 24000 cubic square
B. The surface area = 2(1/2 x 8 x 30) + 2(17 x 50) + 2(50 x 12) + 2(30 x12) + 950 x 30)
= 5360ft square
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A student is running a 10-kilometer race. He runs 1 kilometer every 4 minutes. Select the function that describes his distance from the finish line after x minutes.
f of x equals one fourth times x plus 10
f of x equals negative one tenth times x plus 4
f of x equals one tenth times x plus 4
f of x equals negative one fourth times x plus 10
The distance left after x minutes is modeled by the linear equation:
f(x)= 10 - (1/4)*x
The correct option is the last one.
Which equation models the distance after x minutes?We know that the total distance of the race is 10 kilometers, and the student runs 1 kilometer every 4 minutes, so the student's speed is:
S = 1km/4min = 0.25 km/min
The distance to the finish line is modeled by an equation of the form:
f(x) = total distance - speed*time
Here we know that:
total distance = 10kmspeed = 0.25km/mintime will be our variable, x.Then the equation is:
f(x)= 10 - (0.25)*x
We can rewrite the decimal as a fraction:
f(x)= 10 - (1/4)*x
The correct option is the last one.
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Write an absolute value Inequality for the graph below. Use x for your variable.
The inequality is [tex]|x|\geq4[/tex].
What are the inequalities of a graph?
We can see the areas that satisfy one or more inequalities by visualizing them as inequalities on a graph. These inequalities can be represented using straight-line graphs in GCSE maths because they are frequently linear. You might need to draw lines and designate a region that fulfills the system of inequalities to find solutions to the inequalities, or you might need to use the graphs that have already been provided.
Here, in the graph given, we see that the points on and from 4 / -4 are highlighted. Since the interval has 0 in between, we write the variable as [tex]|x-0|[/tex]. Now, the graph excludes points from -4 to 4.
Therefore, we conclude that the absolute value inequality for the given graph is [tex]|x|\geq4[/tex].
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Absolute value is 7 or greater.
What is inequality?
In mathematics, an inequality is a statement that compares two values or expressions using one of the following symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example:
2 < 3 (2 is less than 3)
5 > 3 (5 is greater than 3)
2 ≤ 3 (2 is less than or equal to 3)
5 ≥ 3 (5 is greater than or equal to 3)
Inequalities can be used to describe a range of values that a variable can take on. For example, the inequality x > 0 describes all values of x that are greater than 0.
Inequalities can be solved by finding the values of the variable that make the inequality true. For example, the solution to the inequality x > 0 is all values of x that are greater than 0, such as 1, 2, 3, etc.
Inequalities can also be combined using the logical operators "and" (represented by the symbol ∧) and "or" (represented by the symbol ∨). For example, the inequality x > 0 ∧ x < 10 describes all values of x that are greater than 0 and less than 10, such as 1, 2, 3, etc. The inequality x > 0 ∨ x < 10 describes all values of x that are either greater than 0 or less than 10, which includes all values of x.
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Last month, Lucy and Britney each read three books. The tables show the number of pages in each book and the time it took Lucy and Britney to read each book. Which of the tables, if any, represent a proportional relationship? Responses
The table that represents a proportional relationship is Table A.
What is a proportional relationship?A proportional relationship exists between two variables when their ratios are equal or equivalent.
With a proportional relationship, there is a constant rate of change between the comparable variables.
Proportions refer to ratios of equality between two or more variables.
Where there is a proportional relationship, we can determine the constant rate of change as the same quotient.
For instance, when the number of pages per book is divided by the reading time, a constant rate of change is established if the reading speed is the same for all three books.
Table A:Pages Lucas Read 208 156 234
Time (hours) 8 6 9
Reading speed 26 26 26 (234/9)
Table B:Pages Britney Read 168 120 348
Time (hours) 6 4 12
Reading speed 28 30 29 (348/12)
Thus, the constant rate of change or proportional relationship can be found in Table A and not in Table B.
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Question Completion:Table A:Pages Lucas Read 208 156 234
Time (hours) 8 6 9
Table B:Pages Britney Read 168 120 348
Time (hours) 6 4 12
Write an equation in slope-intercept form of the line that passes of the line that passes through the given points and is parallel to the graph of the given equation. (2,-2) ; y= -x-2
The equation of the line that passes through the point (2, -2) and is parallel to the graph of y = -x - 2 is y = -x - 4.
What is the system of linear equations?
A system of linear equations is a group of one or more linear equations that can be solved by using a simultaneous equation or elimination method. Therefore, by using the substitution method and writing the solution of the system in terms of coordinate point (x, y) = (-1, 1).
To write the equation of a line in slope-intercept form (y = mx + b), we need to find the slope of the line (m) and the y-intercept (b).
To find the slope of the line, we can use the slope formula:
m = (y2 - y1)/(x2 - x1)
The equation y = -x - 2 is in slope-intercept form, so the slope of the line is -1.
The y-intercept is the point where the line intersects the y-axis. This point is (0, b), where b is the y-intercept.
Since the given line passes through the point (2, -2), we can substitute these values into the slope-intercept form of the equation to find the y-intercept:
-2 = -1 * 2 + b
Solving for b gives us:
b = -4
Therefore, the equation of the line that passes through the point (2, -2) and is parallel to the graph of y = -x - 2 is y = -x - 4.
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NO LINKS!! Please help me with these problems. Exponential.
Answer:
f(x) = 10400·(1/4)^x; table values: x = -1, f(x) = 162.5g(x) = 1.8·2.4^xIncorrect for b < 1.Step-by-step explanation:
You want to write exponential functions through the given points.
1. TableThe exponential function ...
f(x) = a·b^x
will have the values ...
f(0) = af(1) = ab ⇒ b = f(1)/f(0)Using these relations we can write the function from the table values:
f(x) = 10400·(2600/10400)^x
f(x) = 10400·(1/4)^x
Then your table is ...
[tex]\begin{array}{|c|c|c|c|c|c|c|}\cline{1-7}\vphantom{\dfrac{b}{g}}x&-1&0&1&3&4&6\\\cline{1-7}\vphantom{\dfrac{b}{g}}f(x)&41600&10400&2600&162.5&40.625&2.5390625\\\cline{1-7}\end{array}[/tex]
2. PointsUsing the given points in the exponential function form, we have ...
4.32 = a·b^1
59.71968 = a·b^4
Dividing the second equation by the first, we find ...
59.71968/4.32 = b^3 ⇒ b = 2.4
Using this in the first equation, we get ...
4.32 = a·2.4 ⇒ a = 1.8
The equation of the function is ...
g(x) = 1.8·2.4^x
3. IncreasingThe exponential function ...
y = a·b^x . . . . . . a > 0, b > 0
will be increasing when the growth factor (b) is greater than 1. When 'b' is less than 1, the function will be decreasing.
Sarah's belief is incorrect.
Answer:
[tex]\textsf{1.} \quad f(x)=10400(0.25)^x[/tex]
(-1, 41600) and (3, 162.5)
[tex]\textsf{2.} \quad g(x)=1.8(2.4)^x[/tex]
3. Sarah is incorrect.
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$f(x)=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]
Question 1The y-intercept is when x = 0.
Therefore, from inspection of the given table, the y-intercept is 10400:
[tex]\implies a=10400[/tex]
Substitute point (1, 2600) and a = 10400 into the exponential function and solve for b:
[tex]\implies 2600=10400b^1[/tex]
[tex]\implies 2600=10400b[/tex]
[tex]\implies b=\dfrac{2600}{10400}[/tex]
[tex]\implies b=0.25[/tex]
Therefore, the equation of the function is:
[tex]\boxed{f(x)=10400(0.25)^x}[/tex]
Find the value of x when f(x) = 41600:
[tex]\begin{aligned} f(x)&=41600\\\implies 10400(0.25)^x&=41600\\(0.25)^x &=\dfrac{41600}{10400}\\(0.25)^x &=4\\\ln (0.25)^x &=\ln 4\\x \ln (0.25)&=\ln 4\\x&=\dfrac{\ln 4}{\ln (0.25)}\\x&=-1\end{aligned}[/tex]
Find the value of f(x) when x = 3:
[tex]\begin{aligned}x=3 \implies f(3) & =10400(0.25)^3\\& =10400(0.015625)\\& = 162.5\end{aligned}[/tex]
Therefore, the completed table is:
[tex]\begin{array}{|c|c|c|c|c|c|c|}\cline{1-7} \vphantom{\dfrac12}x & -1 & 0 & 1 & 3 & 4 & 6\\\cline{1-7} \vphantom{\dfrac12}f(x) &41600 &10400 &2600 &162.5 &40.625 & 2.5390625\\\cline{1-7} \end{array}[/tex]
Question 2Given points:
(1, 4.32)(4, 59.71968)Substitute the given points into the exponential formula g(x) = abˣ
[tex]\implies ab=4.32[/tex]
[tex]\implies ab^4=59.71968[/tex]
To find b, divide the equations:
[tex]\implies \dfrac{ab^4}{ab}=\dfrac{59.71968}{4.32}[/tex]
[tex]\implies b^3=13.824[/tex]
[tex]\implies b=\sqrt[3]{13.824}[/tex]
[tex]\implies b=2.4[/tex]
Substitute one of the points and the found value of b into the equation and solve for a:
[tex]\implies 2.4a=4.32[/tex]
[tex]\implies a=\dfrac{4.32}{2.4}[/tex]
[tex]\implies a=1.8[/tex]
Therefore, the equation of the function is:
[tex]\boxed{g(x)=1.8(2.4)^x}[/tex]
Question 3Sarah is incorrect. For an exponential function in the form a · bˣ:
If a > 0 and b > 1 then it is an increasing function.If a > 0 and 0 < b < 1 then it is a decreasing function.what is 1/4 divied by 2 and 2/3
Answer:
3/32
Step-by-step explanation:
When dividing fractions, we use keep change flip.
We keep the 1/4, change into multiplication, and flip 2 and 2/3.
First, we want to change 2 and 2/3 into improper fraction form, which is 8/3.
Our equation is now :
1/4 divided by 8/3
Now, we apply keep change flip.
1/4 * 3/8
Multiply numerator by numerator and denominator by denominator.
3/32
3/32 is your final answer. It cannot be simplified further.