The expression for the area of the framed portrait in terms of w is: A = [tex]1.5w^{2}[/tex] + 12w + 64.
How to simplify an expression?Simplifying an expression means to write it in its simplest form by combining like terms and removing any redundant or unnecessary elements. There are several techniques you can use to simplify expressions, including:
Combining like terms: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms. To simplify an expression, you can add or subtract like terms.
Distributing: Distributing means multiplying a constant or variable by each term within a set of parentheses. For example, if you have the expression 2(x + 3), you can distribute the 2 to each term within the parentheses: 2x + 6.
Factoring: Factoring means finding a common factor among a set of terms and grouping them together. For example, if you have the expression [tex]x^{2}[/tex] + 4x, you can factor out the x: x(x + 4).
Simplifying rational expressions: Rational expressions are expressions that have one fractional term. To simplify a rational expression, you can cancel out common factors in the numerator and denominator.
Let's call the width of the portrait w and the height h. We know that the height of the portrait is 1.5 times its width, so h = 1.5w.
The area of the portrait is the product of its width and height: A = w * h = w * 1.5w = [tex]1.5w^{2}[/tex].
The width of the frame is 4 inches all along its perimeter, so the total width of the framed portrait is w + 4 + 4 = w + 8.
The total height of the framed portrait is h + 4 + 4 = h + 8 = 1.5w + 8.
The area of the framed portrait is the product of its width and height: A = (w + 8) * (1.5w + 8) = (w + 8) * 1.5w + 8 * (w + 8) = 1.5w^2 + 12w + 64.
So, the expression for the area of the framed portrait in terms of w is: A = [tex]1.5w^{2}[/tex] + 12w + 64.
This expression is already in standard form.
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You pick a card at random. 4 5 6 7 What is P(4 or less than 6)? Write your answer as a percentag
Answer:
50%
Step-by-step explanation:
Only 4 and 5 can be considered because 6 is not included, so the probability would be 2/4, or 50%.
Answer: 50%
Step-by-step explanation: It says if you pick a random card that is less than 6 or 4. That means that 4 and 5 are the ones you hope to get. 2/4 = 50%.
in a right triangle, a and b are the lengths of the legs and c is the length of the hypotenuse. if a 56 millimeters and c = 65 millimeters, what is b? if necessary, round to the nearest tenth.
Answer:
b = 33 millimeters.
Step-by-step explanation:
In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b). So, using a=56 and c=65, we can find b by solving the equation: 56^2 + b^2 = 65^2. This gives b = sqrt(1089) = 33 millimeters :)
Express 2x² — 12x + 23 in the form a (x + b)² + c.
-
Answer:
Below
Step-by-step explanation:
This is changing the form of the parabola from quadrtatic form to vertex form using the 'complete the square method'
2x^2 - 12x + 23 first, to complete the square the x^2 coefficient must be =1
so factor out 2 form the x components
2 (x^2 -6x) + 23 now....take 1/2 of the 'x' coefficient ( this would be -3)
square it and add this value in the parentheses and subtract it on the outside of the parentheses ( it will be multiplied by the '2' when you need to subtract it....making it -18) )
2 (x^2 - 6x +9 ) - 18 + 23 now reduce the underlined portion
2(x-3)^2 +5 Done .
lucia and ben have saved $150. lucia saved $2 for every $1 that ben saved. how much money did each person save
Lucia saved the total amount of $100 and Ben saved $50. Lucia saved twice as much as Ben so for every $1 Ben saved, Lucia saved $2.
Lucia and Ben have saved a total of $150. To solve this problem, we need to determine how much each person saved. We know that Lucia saved twice as much as Ben, so for every $1 Ben saved, Lucia saved $2. Since they have saved a total of $150, we can set up a proportion to solve for Ben's amount. $2:$1::$100:$x We can solve for x to find Ben's amount saved. $2x = $100, so x = $50. Therefore, Lucia saved $100 and Ben saved $50. Lucia saved twice as much as Ben so for every $1 Ben saved, Lucia saved $2.
The complete question: Together, Lucia and Ben have saved $150. Lucia saved $2 for every $1 that Ben saved. How much money did each person save?
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Subtract -7a^2+3a-9−7a
2
+3a−9minus, 7, a, squared, plus, 3, a, minus, 9 from 5a^2-6a-45a
2
−6a−45, a, squared, minus, 6, a, minus, 4.
Your answer should be a polynomial in standard form.
The required polynomial is [tex]12a^{2} -9a+5[/tex]
What is a polynomial?A polynomial is a expression that contains variables and coefficients, that involves all the four basic arithmetic operations.
Here the given two polynomials are,
[tex]-7a^{2} +3a-9 , 5a^{2} -6a-4[/tex].
subtracting these two polynomials we get,
[tex](5a^{2} -6a-4) - (-7a^{2} +3a-9 ) = 12a^{2} -9a +5[/tex].
Hence, required polynomial is [tex]12^{2} -9a+5.[/tex]
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In proof testing of circuit boards, the probability that anyparticular diode will fail is 0.01.Suppose a circuit board contains 200 diodes. How manydiodes would you expect to fail, and what is the standarddeviation of the number that are expected to fail? What is the (approximate) probability that atleast four diodes will fail on a randomly selected board?If five boards are shipped to a particularcustomer, how likely is it that at least four of them willwork properly? (A board works properly only if all its diodes work)
The standard deviation is 1.482 when the probability of any particular diode is 0.01, and the circuit board is 200 diodes.
A Bernoulli trial is defined as a random experiment with exactly two possible outcomes, that are "success" and "failure", in which the probability of success is the same every time the experiment is conducted.
The probability for the Binomial distribution is given as:
[tex]P(X)=(^{n} C_x)P^x(1-P)^n^-^x[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]^nC_x=\frac{x!}{(n-x)!x!}[/tex]
For this case, we assume that the random variable of interest X= number of circuits that are expected to fail in a sample of 200 selected and we know that:
X simBino (n=200,p=0.01)
The expected value is given by:
E(x)=n*p=200*0.01=2
The variance is given by:
Var(x)np(1-P)=200*0.01*(1-0.01)=1.98
And the deviation is given by:
sd(X)=√1.98-1.407
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assume the speed of vehicles along a stretch of i-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph. what proportion of the vehicles would be going less than 50 mph?
The proportion of vehicles travelling less than 50 mph is 0.1587, as it is the area under the normal curve from 0 to 50 mph.
The speed of vehicles travelling along a stretch of I-10 is assumed to have an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph. This means that the majority of vehicles will travel at speeds close to the mean, with fewer vehicles travelling at speeds significantly above or below the mean. To calculate the proportion of vehicles travelling less than 50 mph, we must calculate the area under the normal curve from 0 mph to 50 mph. This is done using a z-score, which is calculated by subtracting the mean from the value and dividing by the standard deviation. In this case, the z-score for 50 mph is -1.375. The area under the normal curve from 0 mph to -1.375 is 0.1587, which is the proportion of vehicles travelling less than 50 mph.
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A sum of money is divided between Bipana
and Karishma in the ratio of 3: 4. If
Karishma gets Rs 152, how much will
Bipana get?
Ans: Rs 114
Bipana will get an amount of Rs 114, whereas Karishma gets Rs 152.
What is Ratio?The ratio is depicted as a comparison of two quantities of the same unit to determine how much of one quantity is included in the other.
A sum of money is divided between both in the ratio of 3: 4.
Let the amount of money got by Bipana be 3x and the amount of money got by Karishma be 4x.
Since Karishma gets Rs 152.
So 4x = 152
Now, solving for x:
x = 152/4
Apply the division operation, we get:
x = 38
So, the amount of money got by Bipana is: 3(38) = Rs 114
Therefore, Bipana will get an amount of Rs 114.
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Solve the proportion x+10/x+7=x+2/2x-1
Answer:
x = - 12 , x = 2
Step-by-step explanation:
[tex]\frac{x+10}{x+7}[/tex] = [tex]\frac{x+2}{2x-1}[/tex] ( cross- multiply )
(2x - 1)(x + 10) = (x + 2)(x + 7) ← expand both sides using FOIL
2x² + 19x - 10 = x² + 9x + 14 ← subtract x² + 9x + 14 from both sides
x² + 10x - 24 = 0 ← in standard form
(x + 12)(x - 2) = 0 ← in factored form
equate each factor to zero and solve for x
x + 12 = 0 ⇒ x = - 12
x - 2 = 0 ⇒ x = 2
Drew is experimenting on a circuit. He wants to know if the size of the wire used affects battery life. Identify the variables in drew's experiment
independent variable
dependent variable
controlled variables
PLEASE HELP ASAP THANK YOUU SO MUCH
The size of the wire is the dependent variable and the battery life is the independent variable.
What are dependent and independent variables?In any equation, the variables that represent the input and output are referred to as independent and dependent variables, respectively.
Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that Drew is experimenting on a circuit. He wants to know if the size of the wire used affects battery life.
Here, the battery life is the independent variable, while the size of the wire is the dependent variable.
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In the Summer Olympics in Sydney in 2000, the United States won gold silver, and bronze medals. The ratio of gold to silver medals was 5/3. The ratio of silver medals to bronze medals was 8/11. What were the total numbers of gold, silver, and bronze medals won by the United States?
Answer:
97
Step-by-step explanation:
Gold:silver
=5:3
=40:24
silver:bronze
=8:11
=24:33
total number=40+24+33=97
1. a tsunami from krakatau hit jakarta traveling about 60 kilometers per hour. what is the average depth of the water between krakatau and jakarta?
The average depth of the water between krakatau and jakarta is 0.41 kilometers.
We have a function as ,
s= 356√d
where,
s = speed
d = depth of the water
If we have speed of tsunami as 60 kilometers per hour putting the value in the equation,
s= 356√d
60 = 356 √d
√d = 60/356
d = 0.41
Therefore, depth of the water between krakatau and jakarta is 0.41 kilometers.
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complete question:
A volcano erupted on the island of Krakatau, Indonesia. The eruption caused a tsunami(a type of wave) to form and travel into the Indian Ocean and into the Java Sea. The speeds(in kilometers per hour) that a tsunami travels can be modeled by s= 356√d where dis the depth (in kilometers) of the water.1.A tsunami from Krakatau hit Jakarta traveling about 60 kilometers per hour. What is the average depth of the water between Krakatau and Jakarta.
solve the equation. [tex]sqrt(25y^2+40y+16)=5y+4[/tex]
[tex]\sqrt{25y^2+40y+16}=5y+4\implies \stackrel{\textit{squaring both sides}}{25y^2+40y+16=(5y+4)^2} \\\\\\ 25y^2+40y+16=25y^2+40y+16\hspace{5em}\textit{infinitely many solutions}[/tex]
the equation on the left is really the equation on the right in disguise.
for a system of equations, that usually means that one equation is the same as the other, so the graph will be the graph of the second one pancaked over the first one, where they touch each other all the way, and since both lines go to infinity, then they have infintely many solutions.
A pole is supported by two wires, one on each side, going in opposite directions. The wires are 14 feet and 17 feet long. If the wires are to be secured to the ground 22 feet from each other, what angle must the 14-foot long wire make with the ground. Solve using law of cosins and sins
The angle that must the 14-foot long wire make with the ground = 50.6°
Consider the following diagram.
AC represents the pole.
AB represents the supporting wire of length 14 ft and AD is wire of length 17 ft.
The wires are to be secured to the ground 22 feet from each other.
BD = 22 ft
We need to find angle that must the 14-foot long wire make with the ground i.e., angle ACD
Using law of cosines,
c² = a² + b² - 2ab. cos(x)
where x = ∠ABD
c is the opposite side of angle x
a and b are adjacent sides of angle x
here, c = 17, a = 14, and b = 22
Substitute these values in above formula,
17² = 14² + 22² - 2(14)(22). cos(x)
289 = 196 + 484 - 616 cos(x)
-391 = -616 cos(x)
cos(x) = (391/616)
x = arccos(391/616))
x = 50.6°
Therefore, the required angle is 50.6°
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in the circle above, point a is the center and the length of arc bcp is 2 5 of the circumference of the circle. what is the value of x ?
Step 1: Calculate the circumference of the circle.
Circumference = 2πr
Where 'r' is the radius of the circle.
Since we know the center of the circle is point A, we can calculate the radius of the circle by measuring the distance from point A to any point on the circumference.
Let's say we measure the distance from point A to point B.
Radius = AB = x
Step 2: Calculate the circumference of the circle.
Circumference = 2πx
Step 3: Calculate the length of arc bcp.
We know that the length of arc bcp is 2/5 of the circumference of the circle. So,
Length of arc bcp = (2/5) × (2πx)
Length of arc bcp = (4πx)/5
Step 4: Solve for x.
To solve for x, we will rearrange the equation:
(4πx)/5 = Length of arc bcp
5 × (4πx)/5 = 5 × Length of arc bcp
4πx = 5 × Length of arc bcp
4πx = 5 × (2/5) × (2πx)
4πx = 4πx
Since both sides of the equation are equal, x = x.
Therefore, the value of x is equal to itself.
The concept of tangents, sectors, angles, the chord of a circle, and proofs are all included in the circle theorem. All points in a plane that are equally far from a fixed point are located in a circle.
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A die is thrown twice. determine the probability that the sum of the rolls is less than 4 given that at least one of the rules is a 1.
If at least one of the rolls is a 1, then there is a 1/12 chance that the total of the two dice roll will be less than 4.
A dice roll is what?idiom, informal, used to indicate that a situation could result in either a good or negative outcome. It's always a gamble to open a new restaurant. Whether we succeed or fail is up to chance.
Dice formula: What is it?Probability is calculated as follows: 3 out of 36 potential outcomes x desired outcomes = 0.0833. 8.33 percent is the calculated percentage. 7 is also the most frequent outcome when two dice are used. There are also six different ways to do it. The chance in this situation is 6 x 36 = 0.167, or 16.7%.
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3204x7756 to one significant figure
Answer:
20000000 (to 1 s.f.)
Step-by-step explanation:
3204×7756=24850224
=20000000 (to 1 s.f.)
The scatter plot shows the number of hours worked x and the earnings y (in dollars) of a food server
a) The number of hours the server must work to earn $35 is of: 2.5 hours.
b) The earnings for five hours of work are given as follows: $75.8.
c) As the number of hours worked increases, the earnings also increase.
How to find the equation of linear regression?To find the regression equation, also called line of best fit or least squares regression equation, we need to insert the points (x,y) in the calculator. These points are given on a table or in a scatter plot in the problem.
The points for this problem are given as follows:
(1,10), (2, 27), (3, 45), (4, 55) and (5, 78).
Inserting these points into a calculator, the equation is given as follows:
y = 16.4x - 6.2.
Hence the number of hours needed to earn $35 is given as follows:
35 = 16.4x - 6.2
x = 41.2/16.4
x = 2.5 hours.
The earnings for five hours of work are given as follows:
y = 16.4(5) - 6.2
y = $75.8.
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4. Greg plans to purchase a bicycle. Bike A costs $270 and has an expected maintenance cost of $36 per year. Bike B costs $360 and has an expected maintenance cost of $20 per year.
a. Write a rational function for the average yearly cost of Bike A.
b. Write a rational function for the average yearly cost of Bike B.
c. Which bike has the lowest average yearly cost if Greg owns the bike for 5 years? Justify your
answer with numbers and words.
If Greg plans to purchase a bicycle.
a. A rational function for the average yearly cost of Bike A is C(x) = 270 + 36x.
b. A rational function for the average yearly cost of Bike B is C(x) = 360 + 20x.
c. Bike A has the lowest average yearly cost over 5 years.
How to find the rational function?a. The average yearly cost of Bike A can be represented by the rational function:
C(x) = 270 + 36x
where:
x is the number of years that Greg plans to own the bike
270 represents the initial cost of the bike.
b. The average yearly cost of Bike B can be represented by the rational function:
C(x) = 360 + 20x
where:
x is the number of years that Greg plans to own the bike
360 represents the initial cost of the bike.
c. To find the lowest average yearly cost, we need to compare the total cost of owning each bike for 5 years. For Bike A, the total cost after 5 years is:
C(5) = 270 + 36(5) = 450
For Bike B, the total cost after 5 years is:
C(5) = 360 + 20(5) = 460
Therefore, Bike A has the lowest average yearly cost over 5 years, since the total cost of owning it is $450, which is $10 less than the total cost of owning Bike B.
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Which of the following is a correct interpretation of the expression 6-13
The expression 6-13 represents starting at 6 on the number line and moving 13 to the left. The correct answer would be an option (C).
What is Number Line?In math, a number line can be defined as a straight line with numbers arranged at equal segments or intervals throughout. A number line is typically shown horizontally and can be extended indefinitely in any direction.
As we know that the numbers on the number line increase as one move from left to right and decrease on moving from right to left.
Thus, the expression 6-13 represents starting at 6 on the number line and moving 13 to the left.
Hence, the correct answer would be option (C).
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What ratios are equivalent to 6/9
Answer: 2:3
Step-by-step explanation:
from there you can multiply by 2 and find the other ratios
what is 4k^1/2 radical form
Answer:
Below
Step-by-step explanation:
4 [tex]\sqrt[2]{k}[/tex] or just 4 [tex]\sqrt{k}[/tex]
The 1/2 power exponent means square root
if it had been 1/5 then the answer would be 4 [tex]\sqrt[5]{k}[/tex]
HELP ASAP!!!!The Singh family ordered a large pizza with a diameter of 20 inches for dinner. If one of the members of the family ate one eighth of the pizza, how many square inches of pizza are remaining? Use 3.14 for π.
274.75 square inches
39.25 square inches
314 square inches
157 square inches
Answer:
274.75 in²
Step-by-step explanation:
Area of a circle = πr² = (3.14)(20/2)² = 314 in²
(314 in²)(1/8) = 39.25 in²
Pizza remaining = 314 - 39.25 = 274.75 in²
Answer: 274.75
Step-by-step explanation:
Diameter=20
r= 20/2=10
A= π[tex]r^{2}[/tex]
= 3.14*10*10
= 314 square inches
1/8 of 314 = 39.25
So, are of remaining pizza = 314-39.25
= 274.75 square inches
What is the rate of change of Y= -10x+300
Answer: The given equation is Y = -10x + 300.
The rate of change of this equation refers to the rate at which Y changes as x changes. This is the same as the slope of the line represented by the equation.
The equation is in the form of y = mx + b, where m is the slope and b is the y-intercept. So, we can see that the slope of the line is -10. This means that for every one unit increase in x, y decreases by 10 units.
Therefore, the rate of change of Y with respect to X is -10.
The hypotenuse of a right triangle has the length of 50 units, and the longer leg of the triangle has a length of 40 units. What is the length of the projection of the shorter leg on the hypotenuse?
In a right triangle the length of the projection of the shorter leg on the given hypotenuse is equal to 18 units.
In a right triangle ,
length of the hypotenuse = 50 units
length of the longer leg is equal to 40 units
let us consider 'x' be the length of the another leg of right triangle.
Using Pythagoras theorem ,
Hypotenuse² = Base² + Altitude²
⇒ 50² = x² + 40²
⇒ x² = 2500 -1600
⇒ x = 30
Let 'y' be the length of the projection of shorter leg on the hypotenuse
Triangle formed by projected line segment and given right triangle are similar.
Using property of similar triangles corresponding sides are in proportion,
30 / 50 = y / 30
⇒ 50 y = 30 × 30
⇒ y= 900 / 50
⇒ y= 18units
Therefore, the length of the projected leg on hypotenuse of the right triangle is 18 units.
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p to view steps... 9 p 2 − 1 p q − 2 q ÷ 1 − 3 p 3 p − 6 9 p 2 - 1 p q - 2 q ÷ 1 - 3 p 3 p - 6
By applying Algebraic fractions simplifying concept, it can conclude that the simplified expression is - (9p + 3) / q.
Algebraic fractions simplifying can be done by factoring the quantifier and denominator, then dividing by the common factors of the quantifier and denominator.
We have the following algebraic fraction:
((9p² - 1) / (pq - 2q)) / (1 - 3p) / (3p - 6)
To simplify this expression, we will do the following steps:
First, we apply the division rule: (a/b) / (c/d) = ad / bc:
((9p² - 1) / (pq - 2q)) / (1 - 3p) / (3p - 6)
⇔ ((9p² - 1) (3p - 6)) / ((pq - 2q) (1 - 3p))
Then we find the factors of (9p² - 1), that are (3p - 1) (3p + 1):
⇔ ((3p - 1) (3p + 1) (3p - 6)) / ((pq - 2q) (1 - 3p))
Factor other expressions that are not already factored in:
⇔ ((3p - 1) (3p + 1) 3 (p - 2)) / (q (p - 2) (1 - 3p))
Now we cancel the common factor (p - 2), so we have:
⇔ (3 (3p - 1) (3p + 1)) / (-q (3p - 1))
Then we cancel the common factor (3p -1), so we have:
⇔ (3 (3p + 1)) / -q
Finally, we expand the expression:
⇔ - (9p + 3) / q
Thus the simplified expression is - (9p + 3) / q.
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Find the value of k.
a.4√3
b. 8
c. 8√2
d. 16
Answer:
d. 8√2
Step-by-step explanation:
The 2 legs of the right triangle are shown as congruent.
k is the hypotenuse.
k² = 8² + 8² = 128
√k² = k = √128 = √64·2 = 8√2
central africans learned long ago to use observations of the waxing crescent moon (once each month) to predict when the rainy season was coming. use the graph for central africa. in what month(s) does this region get about 200 millimeters of rainfall?
By using the graph, we can determine that Central Africa receives about 200 millimeters of rainfall in the months indicated by the highest points on the graph.
A graph is a visual representation of data that shows the relationship between two or more variables.
The graph shows the amount of rainfall in millimeters that Central Africa receives each month. By looking at the graph, we can see that the highest amount of rainfall is around 200 millimeters. The months in which Central Africa receives approximately 200 millimeters of rainfall are indicated by the highest points on the graph.
To determine the exact month or months in which Central Africa receives about 200 millimeters of rainfall, we need to look at the axis that represents the amount of rainfall. The vertical axis shows the amount of rainfall in millimeters, and the horizontal axis shows the months of the year.
To determine the month or months in which Central Africa receives about 200 millimeters of rainfall, we need to look for the points on the graph that are closest to 200 millimeters on the vertical axis.
These points will correspond to the months in which Central Africa receives about 200 millimeters of rainfall.
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A store sells regular towels for $7 and beach towels for $12. The store sells
42 towels, represented by the equation x + y = 42, where x is the number
of regular towels and y is the number of beach towels. The equation
7x + 12y = 339 represents the total revenue. How many of each type of
towel did the store sell?
The store sold 33 regular towels and 9 beach towels
Here, the store sells 42 towels, represented by the equation
x + y = 42, where x is the number of regular towels
and y is the number of beach towels.
and the equation 7x + 12y = 339 represents the total revenue.
We have system of equations:
x + y = 42 ........(1)
7x + 12y = 339 .........(2)
From equation (1),
x = 42 - y ..........(3)
Substitute above value of x in equation (2),
7(42 - y) + 12y = 339
294 - 7y + 12y = 339
5y = 339 - 294
5y = 45
y = 9
subatitute above value of y in equation (3)
x = 42 - 9
x = 33
Thus, the number of regular towels = 33 and the number of beach towels = 9
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Ryan is trying to start a new colony of ants. He starts with 12 ants and know the number of ants will will triple each day write a function using function notation which can be used to determine the number of ants Ryan will have in x number days.
The function to determine the number of ants Ryan will have in x number of days can be represented as: f(x) = 12 * 3^x
How to calculate the valueIt should be noted that a function shows the relationship between the variables that are provided or the data given regarding an information. In this case, it is vital to note that they have at least two terms which have to be related by through an operator.
The function to determine the number of ants Ryan will have in x number of days can be represented as:
f(x) = 12 * 3^x
where f(x) is the number of ants after x days, and 3^x represents the tripling of the number of ants each day.
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