To prove that removing the root of a k-th order binomial tree results in k binomial trees of smaller orders, we will use mathematical induction.
Base case: For k = 1, removing the root of a 1st order binomial tree results in zero smaller binomial trees, which is true.
Inductive hypothesis: Assume that removing the root of a k-th order binomial tree results in k binomial trees of smaller orders, for some positive integer k.
Inductive step: We want to prove that removing the root of a (k+1)-th order binomial tree results in (k+1) binomial trees of smaller orders.
By definition, a (k+1)-th order binomial tree, denoted by B(k+1), is formed by joining two k-th order binomial trees, denoted by Bk, with one of them as the leftmost child of the root of the other. Let us remove the root of B(k+1) to obtain a new binomial tree, denoted by B'.
Since B(k+1) is formed by joining two k-th order binomial trees, we can apply the inductive hypothesis to each of them to obtain that removing their roots results in k smaller binomial trees each. Let these smaller binomial trees be denoted by B1, B2, ..., Bk and C1, C2, ..., Ck,
respectively.
Now, consider B' and observe that it is a k-th order binomial tree, since it has k children, each of which is a smaller binomial tree. Moreover, B' is obtained from B(k+1) by removing the root, which is the parent of the two k-th order binomial trees Bk and Ck. Thus, we have decomposed B' into k smaller binomial trees, namely B1, B2, ..., Bk and C1, C2, ..., Ck.
Therefore, we have shown that removing the root of a (k+1)-th order binomial tree results in (k+1) binomial trees of smaller orders, which completes the induction.
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Question 12
Identify Structure Choose two different inequalities that each has a solution of x ≤ 4. One inequality should involve multiplication with a
negative coefficient and the other should involve division with a negative coefficient.
□A) 2-2
□ B) 421
943-1
D)-3x ≥ 12
□6)-2x ≤-8
OF)-5x2-20
↓
The equation that would give the solution of x ≤ 4 is -2x ≤-8
What is inequality in mathematics?In mathematics, "inequality" refers to a relationship between two expressions or values that is not equal to each other. Therefore, inequality emerges from a lack of balance.
the equation is given as -2x ≤-8. We are to solve the inequality so as to get the value of x
we are to divide through -2x ≤-8 by -2
-2x / -2 ≤ -8 / -2
x ≤ 4
the other parts of the question are not clearly stated
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rcentage of people who completed or more years of college listed by state are the percentages of the population who have completed or more years of a college education. construct a frequency distribution with classes.
The frequency distribution table of the data is illustrated below.
The data we have been given represents the percentage of people who have completed 4 or more years of college education in different states. To create a frequency distribution, we need to first decide on the number of classes we want to use. In this case, we will use 7 classes.
Next, we need to determine the range of values for each class. To do this, we first find the minimum and maximum values in the data set, which are 18.9 and 37.9, respectively.
The range of values for each class will be (max value - min value) / number of classes, which in this case is (37.9 - 18.9) / 7 = 2.86.
We will start with the first class, which will include all values from 18.9 to 21.76 (the lower limit of the first class plus the range of values for each class). The next class will include values from 21.76 to 24.62, and so on, until we reach the final class which will include all values from 33.18 to 36.04 (the upper limit of the final class plus the range of values for each class).
To create the frequency distribution, we count the number of values in the data set that fall into each class. For example, the first class includes values between 18.9 and 21.76, and there are 2 values in the data set that fall into this range (21.4 and 20.0). We continue this process for each class until we have a table that shows the frequency of values in each class.
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Complete Question:
Percentage of People Who Completed 4 or More Years of College Listed by state are the percentages of the population who have completed 4 or more years of a college education. Construct a frequency distribution with 7 classes.
21.4,26.0,25.3,19.3,29.5,35.0,34.7,26.1,25.8,23.4,27.1,29.2,24.5,29.5,22.1,24,3,28.8,20,0,20.4,26.7,35.2,37.9,24.7,31.0,18.9,24.5,27.0
take your time in 2-6
Which question can be answered using the expression 3 ÷ 1
?
4 8
Responses
A
How many 1 -pound pieces of fudge are in 3
-pound fudge?
8 4How many 1 -pound pieces of fudge are in 3 -pound fudge? 8 4
B
How many 3 -pound pieces of fudge are in 1 -pound fudge?
4 8How many 3 -pound pieces of fudge are in 1 -pound fudge? 4 8
C
Rob ate 1 of 3 pound of fudge. How much fudge did Rob eat?
8 4Rob ate 1 of 3 pound of fudge. How much fudge did Rob eat? 8 4
D
Rob ate 3 of 1 pound of fudge. How much fudge did Rob eat?
4 8
The question that can be answered using the expression 3 ÷ 1 is "How many 1-pound pieces of fudge are in a 3-pound fudge?". The correct option is B.
The expression "3 ÷ 1" is a mathematical expression that represents a division operation. In this case, the operation is "3 divided by 1".
To understand what this expression means, we can think of it in terms of a real-world scenario. For example, we can think of it as dividing 3 pounds of fudge into 1-pound pieces.
If we divide 3 pounds of fudge into 1-pound pieces, we can ask the question "How many 1-pound pieces of fudge are in 3 pounds of fudge?" This is the question that can be answered using the expression "3 ÷ 1".
To solve this division problem, we simply divide 3 by 1. The result is 3. This means that there are 3 one-pound pieces of fudge in 3 pounds of fudge.
So, to summarize, the expression "3 ÷ 1" means "3 divided by 1" and can be interpreted as dividing 3 pounds of fudge into 1-pound pieces. The answer to the question "How many 1-pound pieces of fudge are in 3 pounds of fudge?" is 3.
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I need help with this
Answer: 29 yards
Step-by-step explanation:
First, we want to find how much she spent. We will subtract the leftover from the original amount.
$15 - $9.49 = $5.51 spent
Next, we will convert this value into cents.
$5.51 = 551 cents
Last, we will divide this by the price per yard to find how many yards she bought.
551 / 19 = 29 yards
Answer:
29 yards
Step-by-step explanation:
$15.00 - $9.49= $5.51
$5.51 divide by .19 per yard = 29
In a class of 30 students x +10 study algebra, 10+3 study statistics, 4 study both algebra and statistics. 2x study only Algebra and 3 study neither algebra nor statistics
1. Illustrate the information one a Venn diagram.
2. How many students study;
A. Algebra
B. Only statistics
1. The Venn diagrams are attached
2. When the statistics students number = 10·x + 3, we have;
The number of students that study
Algebra = 128/11Statistic = 213/11When the statistics students number = 2·x + 3, we have;
The number of students that study
Algebra = 16Statistic = 15What is the rationale for the above response?The parameters given are;
Total number of students = 30
Number of students that study algebra n(A) = x + 10
Number of students that study statistics n(B) = 10·x + 3
Number of student that study both algebra and statistics n(A∩B) = 4
Number of student that study only algebra n(A\B) = 2·x
Number of students that study neither algebra or statistics n(A∪B)' = 3
Therefore;
The number of students that study either algebra or statistics = n(A∪B)
From set theory we have;
n(A∪B) = n(A) + n(B) - n(A∩B)
n(A∪B) = 30 - 3 = 27
Therefore, we have;
n(A∪B) = x + 10 + 10·x + 3 - 4 = 27
11·x+13 = 27 + 4 = 31
11·x = 18
x = 18/11
The number of students that study
a. Algebra
n(A) = 18/11 + 10 = 128/11
b. Statistic
n(B) = 213/11
Hence, we have;
n(A - B) = n(A) - n(A∩B) = 128/11 - 4 = 84/11
Similarly, we have;
n(B - A) = n(B) - n(A∩B) = 213/11 - 4 = 169/11
However, assuming n(B) = (2·x + 3), we have;
n(A∪B) = n(A) + n(B) - n(A∩B)
n(A∪B) = 30 - 3 = 27
Therefore, we have;
n(A∪B) = x + 10 + 2·x + 3 - 4 = 27
2·x+3 + x + 10= 27 + 4 = 31
3·x = 18
x = 6
Therefore, the number of students that study
a. Algebra
n(A) = 16
b. Statistics
n(B) = 15
Hence, we have;
n(A - B) = n(A) - n(A∩B) = 16 - 4 = 12
Similarly, we have;
n(B - A) = n(B) - n(A∩B) = 15 - 4 = 11
See the attached Venn Diagrams.
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you are testing h0: μ=10 against ha: μ≠10 based on an srs of 15 observations from a normal population. what values of the t statistic are statistically significant at the α=0.005 level?
The values of the t statistic that are statistically significant at the α=0.005 level are t < -2.977 or t > 2.977.
To determine the values of the t statistic that are statistically significant at the α=0.005 level, we need to find the critical values of the t-distribution with 14 degrees of freedom (df = n - 1 = 15 - 1 = 14).
Using a t-table or a calculator, we can find that the critical values for a two-tailed test with α=0.005 are -2.977 and 2.977. This means that any t statistic less than -2.977 or greater than 2.977 would be considered statistically significant at the α=0.005 level.
In other words, if the calculated t statistic falls within the range of -2.977 to 2.977, we would fail to reject the null hypothesis (H0: μ=10) and conclude that there is not enough evidence to suggest that the population mean is different from 10. However, if the calculated t statistic falls outside of this range, we would reject the null hypothesis and conclude that there is evidence to suggest that the population mean is different from 10.
So, the values of the t statistic that are statistically significant at the α=0.005 level are t < -2.977 or t > 2.977.
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A cable company charges an installation fee to install cable at a residence. There is also a monthly fee for the cable service. The total cost for months 2, 4, 6 and 8 are $145, $255, $365, and $475, respectively. How much is the installation fee? Assume that the relationship between the two quantities is linear
$13,000 and $15,000.
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Fatima has a total of $8 to spend to make fruit smoothies. She will use two types of fruit. The table to the right shows the cost of each type of fruit per cup.
For the mango and pineapple mixture, find a linear equation in standard form that models how much mango and pineapple she can buy, where x is cups of mango and y is cups of pineapple.
Mango=0.50
Pineapple=0.75
Strawberry=1.00
Answer:
0.75y + 0.5x = 8.00
Step-by-step explanation:
question 6 the area of the rectangle is 48x3y5 square inches. its width is 6xy2 inches. what is the length of the rectangle?
The length of the rectangle is [tex]8x^2y^3[/tex] inches when the area of the rectangle is [tex]48x^3y^5[/tex] square inches.
To find the length of the rectangle, we can use the formula for the area of a rectangle, which is:
Area = length x width
We are given that the area of the rectangle is [tex]48x^3y^5[/tex] square inches, and its width is [tex]6xy^2[/tex] inches. Therefore, we can substitute these values into the formula to get:
[tex]48x^3y^5 = length * 6xy^2[/tex]
Simplifying this equation, we can cancel out the common factors of 6x and [tex]y^2[/tex] on both sides to get:
[tex]8x^2y^3 = length[/tex]
Therefore, the length of the rectangle is [tex]8x^2y^3[/tex] inches.
To verify our answer, we can substitute the length and width back into the formula for the area of a rectangle and check if it matches the given area:
Area = length x width = [tex](8x^2y^3) * (6xy^2) = 48x^3y^5[/tex]
Since this matches the given area, we can be confident that our answer is correct. Therefore, the length of the rectangle is [tex]8x^2y^3[/tex] inches.
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which of the following are true statements? select all that apply: 1. a 95% confidence interval is a random interval that contains the true parameter 95% of the time 2. the true parameter is a random value that has 95% chance of falling in the 95% confidence interval 3. i perform a linear regression and get a 95% confidence interval from 0.4 to 0.5. there is a 95% probability that the true parameter is between 0.4 and 0.5. 4. the true parameter (unknown to me) is 0.5. if i sample data and construct a 95% confidence interval, the interval will contain 0.5 95% of the time. answer :
Among the following statements, The true statements are:
Option (1) a 95% confidence interval is a random interval that contains the true parameter 95% of the time, (2)the true parameter is a random value that has 95% chance of falling in the 95% confidence interval, (3) i perform a linear regression and get a 95% confidence interval from 0.4 to 0.5. there is a 95% probability that the true parameter is between 0.4 and 0.5, (4) the true parameter (unknown to me) is 0.
A 95% confidence interval is a random interval that contains the true parameter 95% of the time.
The true parameter is NOT a random value, and it is not correct to say that it has a 95% chance of falling in the 95% confidence interval.
If you perform a linear regression and get a 95% confidence interval from 0.4 to 0.5,
it means that if you repeated the same experiment many times, 95% of the resulting confidence intervals would contain the true parameter.
It is not correct to say that there is a 95% probability that the true parameter is between 0.4 and 0.5 because the true parameter is either in the interval or not.
If the true parameter is 0.5 and you sample data and construct a 95% confidence interval, the interval may or may not contain 0.5.
The 95% confidence interval is a statement about the probability of the interval containing the true parameter, not about the probability of the true parameter being any specific value.
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a knitted hat pattern comes in styles with stitch options and sizes, and there are yarns appropriate for the hat. disregarding size, how many different hats can be made?
A knitted hat pattern comes in 4 styles with 4 stitch options and 3 sizes, and there are 7 yarns appropriate for the hat. There are 112 different hats that can be made using the knitted hat pattern, disregarding size.
To find the total number of different hats that can be made using the knitted hat pattern, we need to multiply the number of choices for each feature.
There are 4 styles to choose from and 4 stitch options, so there are 4 x 4 = 16 different combinations of style and stitch.
For each of these style and stitch combinations, there are 7 yarns that can be used, so the number of possible yarn choices is 7.
Therefore, the total number of different hats that can be made, disregarding size, is:
16 (style and stitch combinations) x 7 (yarn choices) = 112
This calculation shows how the multiplication principle of counting can be used to find the total number of outcomes in a multi-stage process where the number of choices at each stage is known. In this case, we multiplied the number of style and stitch combinations by the number of yarn choices to find the total number of different hats that can be made.
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Complete question is:
A knitted hat pattern comes in 4 styles with 4 stitch options and 3 sizes, and there are 7 yarns appropriate for the hat. Disregarding size, how many different hats can be made?
A farmer has 7 3/4 rows of radishes in one field,4 3/4 rows of radishes in another field,and 6 1/4 rows of radishes in a third field .How many rows of radishes does he have altogether.
Answer:
18 and 3/4 rows of radishes
Step-by-step explanation:
6 + 4 + 7 = 17
3/4 + 3/4 + 1/4 = 7/4
7/4 = 1 3/4
17 + 1 3/4 = 18 3/4
Answer: 18 and 3/4 rows of radishes
Since all the fractions have the same denominator,
you can combine 3/4 + 3/4 + 1/4 together into 1 and 3/4
Adding in the numbers 7 + 4+ 6 and the 1 and 3/4 would amount to 18 and 3/4
A fruit seller bought a crate of 190 kiwi fruits for $66.50. If he sells each kiwi fruit for 55 cents, what is the
least number of kiwi fruits he must sell in order to make a profit of not less than $20?
I NEED HELP PLEASEEEEEEEEE!!!!!!
Step-by-step explanation:
so, it starts (year 0 after 2000) with 350 foxes.
after 1 year the number had increased by 5%.
that means : 350×1.05
because 5% is represented by 0.05.
5% of 350 is 350×0.05 (= 17.5)
and to add 5% means that this is added to the original 100% (= 1).
100% + 5% = 105%
means
1 + 0.05 = 1.05
so, again, after the first year we have
350×1.05
foxes.
after the second year we have to multiply that by 1.05 again :
350×1.05×1.05
and so on.
so, what we are doing is multiplying the original number of foxes by powers of 1.05 (with the exponent being the number of years after 2000).
f(x) = 350 × (1.05)^x
A small publishing company is releasing a new book. The production costs will include a one-time fixed cost for editing and an additional cost for each book
printed. The total production cost C (in dollars) is given by the function C=15. 95N+750, where N is the number of books.
The total revenue earned (in dollars) from selling the books is given by the function R-32. 80N.
Let P be the profit made (in dollars). Write an equation relating P to N. Simplify your answer as much as possible.
0
?
If The total revenue earned (in dollars) from selling the books is given by the function R-32. 80N.The equation relating the profit P (in dollars) to the number of books N is P = 16.85N - 750.
The profit made by the company is the difference between the total revenue earned and the total production cost, so we can write:
P = R - C
Substituting the expressions for R and C, we get:
P = 32.80N - (15.95N + 750)
Simplifying the right-hand side, we get:
P = 16.85N - 750
This equation shows that the profit made by the company is a linear function of the number of books printed. The slope of the line is the revenue per book (32.80 dollars), minus the cost per book (15.95 dollars), which is 16.85 dollars.
The intercept of the line is the fixed cost for editing (750 dollars). The equation can be used to estimate the profit for any given number of books printed, and to determine the break-even point, which is the number of books that need to be sold to cover the total production cost.
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The school that Stefan goes to is selling tickets to a choral performance. On the first day of ticket sales, the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets. Find the price of a senior citizen ticket and the price of a child ticket.
The school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets.The price of a senior citizen ticket is $17.33 and the price of a child ticket is $5.01.
What is Word Problem?
A word problem is a mathematical problem that is presented in the form of a story or scenario, often using words and symbols to describe a real-world situation. Word problems require the solver to read and understand the problem, identify the relevant information, translate the information into mathematical expressions or equations, and then solve for the unknown quantity.
Word problems can involve a wide range of mathematical concepts, including arithmetic, algebra, geometry, statistics, and calculus, and are often used to test students' problem-solving abilities and mathematical reasoning skills.
Let's call the price of a senior citizen ticket "s" and the price of a child ticket "c".
From the information given, we know:
3s + c = 38 (equation 1)
3s = 52 (equation 2)
Using equation 2, we can solve for s:
3s = 52
s = 17.33 (rounded to two decimal places)
Now we can substitute this value of s into equation 1 and solve for c:
3s + c = 38
3(17.33) + c = 38
c = 38 - 3(17.33)
c = 5.01 (rounded to two decimal places)
So the price of a senior citizen ticket is $17.33 and the price of a child ticket is $5.01.
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3. PLEASE THIS IS URGENT
B. Bill's Bikes sells new and used bikes. Bill buys used bikes, fixes them, and marks them up by 80%. The sales-
person selling the bike receives a 25% commission on the markup.
a. Find the missing values in the table. Show your work below
Strategies include:
Using proportions
Percent bar models
Multiplying with the decimal (or fractional) equivalent
Reasoning about with benchmark percent values
Costs and Revenue for Roberto's Sales
Buying
Markup
Selling
Commission
Profit (money the
(25% of markup) shop makes on the sale)
Price (80% of buying price) Price
$100
$180
$20
$60
$10
$55
$125
PLEASE AND I NEED THE WORK SHOWN FOR BOTH CHARTS
The values that van be put in the table based on the information will be:
Buying price (dollars) 100.
Mark-up 80.
Selling price 180
Commission 20
Profit 60
Buying price (dollars) 10
Mark-up 8
Selling price 18
Commission 2
Profit 6
Buying price (dollars) 55
Mark-up 44
Selling price 99
Commission 11
Profit 33
Buying price (dollars) 125
Mark-up 100
Selling price 225
Commission 25
Profit 75
How to explain the valueIt should be noted that the markup is 80% of the buying price. For example, when the buying price is $10, the markup will be:
= 80% × $10
= $8
The selling price is that addition of buying price and markup. This will be:
= $10 + $8
= $18
The commission is 25% of markup. This will be:
= 25% × 8
= $2
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A 90° rotation around the origin followed by a reflection across the line x = 3 maps △JKL, with coordinates J(–2, 2), K(–2, 6), and L(–8, 2), to △PQR. Find the area of △PQR.
Answer: A 90° rotation around the origin maps point (x, y) to (-y, x). So, the image of point J after a 90° rotation is J' = (-2, 2) -> (-2, -2).
A reflection across the line x = 3 maps point (x, y) to (2x - 3, y). So, the image of point J' after a reflection across x = 3 is J'' = (-2, -2) -> (4, -2).
Similarly, we can find the images of K and L:
K' = (-2, 6) -> (-6, -2)
K'' = (-6, -2) -> (-2, -2)
L' = (-8, 2) -> (2, -8)
L'' = (2, -8) -> (8, -2)
Therefore, the image of triangle JKL after the 90° rotation and reflection is triangle PQR, with vertices P = J'', Q = K'', and R = L''.
The area of triangle PQR can be found using the Shoelace Formula:
P = [(4 -2) + (-2 -2) + (8 -2) * (-2 - (-8))]/2 = [6 + 4 * 6]/2 = 30/2 = 15 square units.
So, the area of triangle PQR is 15 square units.
Step-by-step explanation:
What information is in the text but not in the graph?
Answer:
B. More money was given than expected in 1994
Step-by-step explanation:
Nowhere is it mentioned what amount was expected as donations. Only actual amounts received are shown on the graph
Hence this is an assumption and not necessarily fact
Richard and Teo have a combined age of 28. Richard is 10 years older than twice Teo's age. How old are Richard and Teo?
The algebra word problem method shows that Richard is 22 years old and Teo is 6 years old.
Algebra word problems in mathematicsAlgebra is used in our daily lives without us even recognizing it. Students can relate to and comprehend the significance of mathematics in the real world by solving algebraic word problems in worksheets.
Algebraic word problems usually comprise interpreting words into numbers, variables, their coefficients, and the use of appropriate arithmetic operations.
From the information given, we can the following system of equations as follows:
r + t = 28 --- (1)
r = 2t + 10 --- (2)
(2t + 10) + t = 28
3t + 10 = 28
3t = 28 - 10
3t = 18
t = 18/3
t = 6
From (1), r + t = 28
r + 6 = 28
r = 28 - 6
r = 22
Therefore, we can conclude that Richard is 22 years old and Teo is 6 years old.
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. Solve for n.
scale: 1 1/2 inches: 250 miles
scale measure: 3/4 inches
actual measure: n miles
A. 100 miles
B. 125 miles
C. 187 1/2 miles
D. 281 1/4 miles
Answer:
C) 187 1/2 miles
Step-by-step explanation:
We can start by using the formula for converting between scale measure and actual measure:
actual measure = scale measure * (actual distance / scale distance)
Here, the scale measure is 3/4 inch and the scale distance is 1 1/2 inches, so:
actual measure = (3/4) * (n / 250)
Rearranging the equation, we can isolate n:
n = (actual measure * 250) / (3/4)
Substituting in the actual measure of 3/4 inch, we find:
n = (3/4 * 250) / (3/4) = 250 miles
So the answer is C) 187 1/2 miles.
Simplify: √-363
○ -11√/3
11i
33i
○ 11i√√3
Answer:
11i√√3
Step-by-step explanation:
A sea horse weighs 42oz. Together, 3 sea monkeys and a sea horse weighs the same as 8 sea monkeys plus 9oz. How much does a sea monkey weigh?
Answer:
a sea monkey weighs 6.6oz
Step-by-step explanation:
The population of a large US city is 1,703,210. Show how you could express this in Scientific Notation
The population of a large US city is 1.70321 × 10⁶ in Scientific Notation.
Scientific NotationIt may be referred to as scientific form or standard index form.
Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form.
We know that, a number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.
According to the question,
the population of a large US city is 1,703,210. we need to show this number in Scientific Notation.
The number is written in two parts,
The digits, with the decimal point placed after the first digit, followed by × 10 to a power that puts the decimal point where it should be
Therefore,
1,703,210 = 1.70321 × 10⁶.
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jack's favorite number is 836. jill subtracts a secret number from 836 and her answer is the smallest possible combination of all the digits from jack's favorite number. what is the secret number?
The secret number is 564 as we solve it by given question.
jack's favorite number is 836 so here we have number 836 and
jill subtracts a secret number from 836 and her answer is the smallest possible combination of all the digits from jack's favorite number
so here we have a number 836
let we subtract x from 836 to get a number which is smallest number by using 836
the number which we can make by using these 3 degits are
368, 386, 638, 683, 836, 863 anf from all these numbers the smallest number which can be using 836 is 368
as given in question we have 836 and we will subtract x and get 368 so
836 - x = 368
x = 836 - 368
x = 564
the secrets number is 564.
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HELP pls will mark you the brainliest
The function among all options is y=30.5(0.7)ˣ.
What is the graph of a function?The graph of a function represents a set of all points in a plane. It helps to derive the nature of the function.
The function must satisfy the points that are given in the graph.
When x=0, y=21.35(0.7)ˣ=21.35(0.7)⁰=21.35.
But the given point is (0,30.5).
Hence, it cannot be the required function.
When x=0, y=30.5(21.35)ˣ=30.5(21.35)⁰=30.5.
When x=1, y=30.5(21.35)ˣ=30.5(21.35)¹=651.175.
But the given point is (1,21.35).
Hence, it cannot be the required function.
When x=0, y=30.5(1.3)ˣ=30.5(1.3)⁰=30.5.
When x=1, y=30.5(1.3)ˣ=30.5(1.3)¹=39.65
But the given point is (1,21.35).
Hence, it cannot be the required function.
When x=0, y=30.5(0.7)ˣ=30.5(0.7)⁰=30.5.
When x=1, y=30.5(0.7)ˣ=30.5(0.7)¹=21.35.
When x=2, y=30.5(0.7)ˣ=30.5(0.7)²=14.945≈14.95.
It satisfies all three given points.
Therefore, the required function is y=30.5(0.7)ˣ.
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A man and a woman share a cash prize of £2855 between them in the
ratio 1:4 respectively. The woman shares her winnings between herself,
her daughter and her son in the ratio 5:1:2. How much does her
daughter receive?
Answer:
285.50
Step-by-step explanation:
The total for the ratio of 1:4 is 5. The ratio to the woman to the total is
4:5 or
[tex]\frac{4}{5}[/tex] = [tex]\frac{x}{2855}[/tex] Cross multiply and solve for x
4(2855) = 5x
11420 = 5x Divide both sides by 5
2284 =x
The woman's portion of the total is 2284.
The woman takes the 2284 and divides it in a ratio of 5:1:2. The total is 8.
The daughter will receive the ratio of 1:8 of the total.
[tex]\frac{1}{8}[/tex] = [tex]\frac{x}{2284}[/tex] cross multiply and solve for x
8x = 2284 Divide both sides by 8
x = 285.50
The daughter's portion is 285.50
(15, 2); perpendicular to 5x - y = 2
(a) Write the equation of the line in slope-intercept form.
(b) Write the equation of the line in standard form.
Answer:
see explanation
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given
5x - y = 2 ( subtract 5x from both sides )
- y = - 5x + 2 ( multiply through by - 1 )
y = 5x - 2 ← in slope- intercept form
with slope m = 5
(a)
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{5}[/tex] , then
y = - [tex]\frac{1}{5}[/tex] x + c ← is the partial equation
to find c substitute (15, 2 ) into the partial equation
2 = - [tex]\frac{1}{5}[/tex] (15) + c = - 3 + c ( add 3 to both sides )
5 = c
y = - [tex]\frac{1}{5}[/tex] x + 5 ← in slope- intercept form
(b)
the equation of a line in standard form is
Ax + By = C ( A is a positive integer and B, C are integers )
y = - [tex]\frac{1}{5}[/tex] x + 5 ( multiply through by 5 to clear the fraction )
5y = - x + 25 ( add x to both sides )
x + 5y = 25 ← in standard form
F.E.O.L. Perpendicular to a Given Line
Write the slope-intercept form of the equation of the line described.
1) through: (-2,-3), perp. to y=-x
5
3) through: (2,-2), perp. to y=-
2) through: (5,-2), perp. to y = 5x +4
4) through: (-4,-3), perp. to y=-x+2
The equation of the line are;
a. y = 5/2x + 2
b. y = -1/5x - 1
c. y = -5/2x - 3
d. y = -x - 7
What is the slope intercept form of the linea.
The line passes through point (-2, -3) and it is perpendicular to y = -2/5x
The equation of the line is y = 5/2x + 2
b.
The line passes through the point (5, -2) and it is perpendicular to y = 5x + 4
The equation of the line is y = -1/5x - 1
c.
The line passes through (-2, 2) and it is perpendicular to y = 2/5x - 4
Th equation of the line is y = -5/2x - 3
d.
The line passes through (-4, -3) and it is perpendicular to y = -x + 2
The equation of line is y = -x - 7
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