Answers
Answer:
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Step-by-step explanation:
Related Questions
At a meeting for math enthusiasts, two entrance fees are charged. Anyone who knows the code word “algemath” only pays 2 euros. Those who are unlucky and do not know the code word pay three euros more. You know that 7338 people were present and that the cashier received 17487 euros. How many people knew the password?
Answers
Number of People who knows the Code are 6401
What is Linear Equation in Two Variable?
A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.
Solution:
Let, Number of People who kows the Code be x
Number of Peoplee who do not know the Code be y
According to the Question:
x + y = 7338 ----------(i)
2x + 5y = 17487 ----------(ii)
Multiplying Equation (i) by 2
2x + 2y = 14676 ---------(iii)
Substracting Equation (iii) from Equation (ii)
3y = 2811
y = 937
This means x = 6401
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Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment. If his students performed the experiment 6 1/2 times, how much baking soda did they use?
Answers
Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment. If his students performed the experiment 6 1/2 times, he used 19/4 baking soda.
Define subtraction.To subtract in mathematics is to take something away from a group or a number of objects. The group's total number of items decreases or becomes lower when we subtract from it. The components of a subtraction issue are the minuend, subtrahend, and difference.
Given,
Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment.
Converting mixed fraction to improper function
1 3/4 = 7/4
If his students performed the experiment 6 1/2 times,
Converting mixed fraction to improper function
6 1/2 = 13/2
Subtracting,
7/4 - 13/2
7/4 - 26/4
19/4
Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment. If his students performed the experiment 6 1/2 times, he used 19/4 baking soda.
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The following expressions (-9a3b2 + 13ab5) and (-10ab5 + 12a3b2) represents the length and width of the pizza box respectfully. Find the perimeter of the pizza box. Express the perimeter of the pizza box in the order of descending exponents of a. Please use the palette below to input your answer.
Answers
The perimeter of the box has been obtained as 6a^3b^2 + 6ab^5.
What is the perimeter of the box?We know that a box has the shape of a rectangle thus we would have to apply the formula for the area of a rectangle in this case and we would have;
P = 2(l + b)
P = perimeter
l =length
b = breadth
Then we have;
2[(-9a^3b^2 + 13ab^5) + (-10ab^5 + 12a^3b^2)]
Collect the like terms
2[(-9a^3b^2) + 12a^3b^2 + 13ab^5 +(-10ab^5)]
2[3a^3b^2 + 3 ab^5]
6a^3b^2 + 6ab^5
In order of descending exponents of a, the perimeter is 6a^3b^2 + 6ab^5.
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Marquis sold gift-wrapping paper for a school fundraiser. He sold at least 15 rolls of paper. Write an inequality to represnet the amount of money, d, Marquis earned for the fund-raiser
Answers
The inequality to represent the amount of money, Marquis earned for the fund-raiser is d ≥ 15p
What is the inequality to represent the information?
Inequalities are created through the connection of two expressions. It should be noted that the expressions in an inequality aren't always equal. Inequalities implies that the expressions are not equal. They are denoted by the symbols ≥ < > ≤.
Let the cost of each paper be p.
Since Marquis sold gift-wrapping paper for a school fundraiser and he sold at least 15 rolls of paper.
An inequality to represent the amount of money, Marquis earned for the fund-raiser is:
d ≥ (15 × p)
d ≥ 15p
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a quiz consists of 6 multiple-choice questions with 4 possible responses to each one. in how many different ways can the quiz be answered? (see example 7.)
Answers
His chances of correctly predicting an answer are 1/4.
He has a 3/4 chance of getting an answer incorrectly.
Binomial probability is defined.A random variable is distributed binomially with parameters n, p more precisely if
It is the number of successes over n separate* trials, each of which has a chance of success of p.
What distinguishes possibility from probability?Technically, the answer to whether an occurrence is possible is always "yes" or "no," or 1 or 0. Probability is the measure of how probable an event is to occur given the circumstances, assuming that it is feasible.
For instance, the likelihood of the number 5 appearing when we roll the dice is "Yes" (or 1), but the likelihood of the number 8 appearing is "No" (or 0).
Case 1: Guessing exactly 3 correct:
Choosing exactly 3 to get correct = Choosing exactly
3 to get wrong = 6C3 = (6*5*4)/(3*2*1) = 20 ways.
Guess the wrong answer for the first wrong one 3 ways.
Guess the wrong answer for the second wrong one 3 ways.
Guess the wrong answer for the third wrong one 3 ways.
That's 20*3*3*3 = 540 ways
Case 2: Guessing exactly 4 correct:
Choosing exactly 4 to get correct = Choosing exactly
2 to get wrong = 6C2 = (6*5)/(2*1) = 15 ways.
Guess the wrong answer for the first wrong one 3 ways.
Guess the wrong answer for the second wrong one 3 ways.
That's 15*3*3 = 135 ways
Case 3: Guessing exactly 5 correct:
Choosing exactly 5 to get correct = Choosing exactly
1 wrong = 6C1 = 6 ways.
Guess the wrong answer for the wrong one 3 ways.
That's 6*3 = 18 ways
Case 4: Guessing all 6 correct
There is only 1 way to guess them all correct
Total number of ways to pass: 540+135+18+1 = 694 ways to pass
Total number of ways to answer:
(4 ways to answer question 1) times
(4 ways to answer question 2) times
(4 ways to answer question 3) times
(4 ways to answer question 4) times
(4 ways to answer question 5) times
(4 ways to answer question 6) equals 4*4*4*4*4*4 = 46 = 4096
Probability of passing = 694/4096 = 347/2048 = 0.1694335938
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segments in circles maze
Answers
Answer:
x = 6
Step-by-step explanation:
You want to know the value of x given a tangent to a circle from an external point has length x, while a secant from that point has length x-2 to the first intersection with the circle, and an additional 5 to the second intersection.
Secant/Tangent relationThe product of the segments of the secant to the two points of intersection with the circle is equal to the square of the tangent segment.
(x -2)(x -2 +5) = (x)² . . . . . use given values
x² +x -6 = x² . . . . . . simplify
x -6 = 0 . . . . . . subtract x²
x = 6 . . . . . . add 6
__
Additional comment
There are rules for two secants, for a tangent and secant (as here), and for two chords internal to the circle. These can be easier to remember if you generalize them to a single rule: the product of the distance from the common point to the first circle intersection and the distance to the second circle intersection is the same for each segment.
For a tangent, the two circle intersections are the same point, hence the distance is squared. For chords, the two circle intersections are at the ends of the chords, and the common point is where the chords cross.
Which of the following are factor pairs for 54?select all that apply.
Answers
Answer:
The Pair Factors of 54 are (1, 54), (2, 27), (3, 18), and (6, 9) and its Prime Factors are 1, 2, 3, 6, 9, 18, 27, 54.
Step-by-step explanation:
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The Pair Factors of 54 are (1, 54), (2, 27), (3, 18), and (6, 9) and its Prime Factors are 1, 2, 3, 6, 9, 18, 27, 54.
Here, we have,
The factors of 54 are the numbers, that can divide 54 completely or evenly. When a pair of factors are multiplied together to produce the 54, then they are said to be pair factors. The factors divide the number completely. Hence, these factors cannot be a fraction.
Factors of 54: 1,2,3,6,9,18,27 and 54
Factor pairs of the number 54 are the whole numbers which are not a fraction or decimal number.
To find the factors of a number, 54, we will use the factorization method.
To find factors of 54, we have to divide 54 by all natural numbers from 1 to 54.
54 ÷ 1 = 54
54 ÷ 2 = 27
54 ÷ 3 = 18
54 ÷ 6 = 9
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Point A is located at (5, 6). Find the coordinates of the image A' after the
point is reflected across the x-axis.
(5,-6)
(-5,6)
(-5, -6)
(5,6)
Answers
Answer:
Step-by-step explanation:
Point A = (5, 6) is reflected across the x axis which is y =0.
When the point is reflected across the x axis, the x coordinate
remains positive, but the y coordinate becomes negative.
The 6 becomes negative (-6).
So the coordinates of A' are (5, -6) which is the first answer.
Student grades on a chemistry exam were: 77, 78, 75, 81, 89, 53; 79, 82, 84; 91 a. Select the stem-and-leaf plot of the data. Stem Leaf 5 3 6 7 5 7 8 9
8 1 2 4 9 9 1 Stem Leaf 50 53 60 70 75 77 78 79 80 81 82 84 89 90 91 Stem Leaf 5 53 6
7 77 78 75 79 8 81 89 82 84 9 91 Stem Leaf 5 3 6 7 7859 8 1924 9 1
Answers
The correct stem-and-leaf plot for the data is:
i. Stem: 5,6,7,8,9 Leaf: 3, , 5 7 8 9, 1 2 4 9, 1
This plot shows the distribution of the data values, with the stems representing the tens place and the leaves representing the ones place. For example, the value 53 is represented by the stem 5 and the leaf 3, and the value 82 is represented by the stem 8 and the leaf 2.
The other three stem-and-leaf plots are incorrect because they do not accurately represent the distribution of the data.
A stem-and-leaf plot is a graphical representation of data that shows the distribution of values. The stems represent the tens place and the leaves represent the ones place. It is useful for identifying patterns and trends in the data.
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Find a logarithmic equation that relates y and x. (Round any numeric values to 3 decimal places.)
ln(y)=
x y
1 1.250
2 1.051
3 0.950
4 0.884
5 0.836
6 0.799
Answers
Answer:
[tex]\ln (y) = -0.250 \ln (x)+0.223[/tex]
Step-by-step explanation:
Given table:
[tex]\begin{array}{|c|c|}\cline{1-2} \vphantom{\dfrac12}x & y\\\cline{1-2} \vphantom{\dfrac12} 1 & 1.250\\\cline{1-2} \vphantom{\dfrac12} 2 & 1.051\\\cline{1-2} \vphantom{\dfrac12} 3& 0.950\\\cline{1-2} \vphantom{\dfrac12} 4& 0.884\\\cline{1-2} \vphantom{\dfrac12} 5& 0.836\\\cline{1-2} \vphantom{\dfrac12} 6& 0.799\\\cline{1-2} \end{array}[/tex]
To convert y = axⁿ to linear form, take natural logs of both sides and rearrange:
[tex]\begin{aligned}y=ax^n \implies \ln y &= \ln ax^n\\\implies \ln y &= \ln a + \ln x^n\\\implies \ln y &= \ln a + n \ln x\\\implies \ln y &=n \ln x+ \ln a\end{aligned}[/tex]
This is in the straight-line form y = mx + c.
Substitute the first values of x and y from the table into the natural log formula and solve for ln(a):
[tex]\implies \ln 1.250= n \ln 1+\ln a[/tex]
[tex]\implies \ln 1.250= n (0)+\ln a[/tex]
[tex]\implies \ln a = \ln 1.250[/tex]
[tex]\implies \ln a = 0.223\; \sf (3 \; d.p.)[/tex]
Substitute the found value of ln(a) and the last values of x and y from the table into the natural log formula and solve for n:
[tex]\implies \ln 0.799= n \ln 6+\ln 1.250[/tex]
[tex]\implies n \ln 6=\ln 0.799-\ln 1.250[/tex]
[tex]\implies n =\dfrac{\ln 0.799-\ln 1.250}{\ln 6}[/tex]
[tex]\implies n =-0.250\;\sf (3 \; d.p.)[/tex]
Substitute the found values of n and ln(a) into the formula to create an equation for ln(y):
[tex]\boxed{\ln (y) = -0.250 \ln (x)+0.223}[/tex]
Select all the expressions that are equivalent to 8-12-(6+4).
(Select all that apply.)
8-(6+4)-12
(6+4)-8-12
8-12-6-4
8-6-12+4
Answers
Answer:
8-(6+4)-12
8-12-6-4
Step-by-step explanation:
8-(6+4)-12 and 8-12-6-4 are equivalent expressions because they both compute to -6. This is because in both expressions, the 6+4 is calculated first, yielding 10. Then, the 8 is subtracted from 10, yielding -2. Finally, the 12 is subtracted from -2, yielding -6.
6+4)-8-12 is not an equivalent expression because it does not compute to -6. In this expression, the 6+4 is calculated first, yielding 10. Then, the 8 is subtracted from 10, yielding 2. Finally, the 12 is subtracted from 2, yielding -10.
8-6-12+4 is not an equivalent expression because it does not compute to -6. In this expression, the 8 is subtracted from 6, yielding -2. Then, the 12 is subtracted from -2, yielding -14. Finally, 4 is added to -14, yielding -10.
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Assume that the following statement, S_k, is true:
A factor of (2^(2k-1) + 3^(2k-1)))is 5.
Show that for each integer k ≥ 1, if S_k is true, then S_(k+1) is true.
2^2(k + 1) - 1)) + 3^2(k + 1) - 1)) = 2^(2k + ____) - 1)) + 3^2k + ___) - 1))
= 2^(2k - 1) * 2^(___) + 3^(2k - 1) * 3^(___)
= ______ * 2^(2k-1) + ______ * 3^(2k-1)
= (2^(2k-1) + 3^(2k-1)) + (2^(2k-1) + 3^(2k-1)) + (2^(2k-1) + (3^(2k-1) + ______*3^(2k-1)
Hence, S_k+1 is true or false, which completes the inductive step and the proof by the mathematical induction.
Answers
It is assumed the statement is true for [tex]S_k[/tex]:
A factor of [tex]2^{2k-1}+3^{2k-1}[/tex] is 5.Show that [tex]S_{k+1}[/tex] is also divisible by 5:
[tex]2^{2(k+1)-1}+3^{2(k+1)-1}=[/tex][tex]2^{2k+2-1}+3^{2k+2-1}=[/tex][tex]2^{(2k-1)+2}+3^{(2k-1)+2}=[/tex][tex]2^{2k-1}*2^2+3^{2k-1}*3^2=[/tex][tex]4*2^{2k-1}+9*3^{2k-1}=[/tex][tex]4*2^{2k-1}+4*3^{2k-1}+5*3^{2k-1}=[/tex][tex]4(2^{2k-1}+3^{2k-1})+5*3^{2k-1}[/tex]The first part is divisible by 5 as we assumed in the beginning and the second part has a factor of 5 so the sum is divisible by 5.
Hence it makes [tex]S_{k+1}[/tex] true, so the proof is complete.
All changes Given: A(0,6), B(6,6), C(6,0), D(0,0); ZDAB, LABC, ZBCD, LCDA are right angles. Prove: ABCD is a square.
Answers
Since ABCD has four equal angles and four equal sides, then it is a square.
How to find the distance between two coordinates?The formula for the distance between two coordinates (x₁, y₁) and (x₂, y₂) is; Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
We are given the coordinates;
A(0, 6)
B(6, 6)
C(6, 0)
D(0, 0)
Thus;
AB = √[(6 - 0)² + (6 - 6)²]
AB = 6
BC = √[(6 - 6)² + (0 - 6)²]
BC = 6
CD = √[(0 - 0)² + (0 - 6)²]
CD = 6
AD = √[(0 - 0)² + (0 - 6)²]
AD = 6
Thus, we conclude that all the sides are equal and it is a square.
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please help me i will give brainliest thank you so much
Answers
The problem can be made into the equation 8X + 50 = 130, with X being the hours worked. This can be solved like this:
8X + 50 = 130
8X = 80
X = 10
Hope this helps!
Caroline baked cookies from a recipe that called for 3/4 cups of sugar she planned to triple the recipe how much sugar did she need?
Answers
The amount of sugar needed for the recipe to be tripled is 9 / 4 cups.
How to find the amount of sugar needed?Caroline baked cookies from a recipe that called for 3/4 cups of sugar. she planned to triple the recipe. Therefore, the amount of sugar she needs is calculated as follows:
1 recipe = 3 / 4 cups of sugar
She wants to triple the recipe.
Hence,
1 recipe = 3 / 4 cups of sugar
3 recipe = ?
cross multiply
amount of sugar needed = 3 / 4 × 3
amount of sugar needed = 9 / 4
amount of sugar needed = 9 / 4 cups of sugars
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A concession stand sells hot dogs for $2 and cheeseburgers for $3. One day, 486 items were sold and the owners earned $1218.
How many hot dogs and cheeseburgers were sold?
Answers
The Total Number of Hot Dogs sold were 240 and Cheese Burgers sold were 246
What is Linear Equation in Two Variables?
A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.
Solution:
Let, Number of Hot Dogs Sold = x
Number of Cheese Burgers Sold = y
According to the Question:
x + y = 486 ----------(i)
2x + 3y = 1218 ----------(ii)
Multiplying Equation (i) by 2
2x + 2y = 972 ----------(iii)
Substracting Equation (iii) from Equation (ii)
y = 246
x = 240
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A function is shown: f(x)= 2/3x+3
What is the value of f(18)?
Answers
Answer:
15
Step-by-step explanation:
When you see something like f (18), this means 18 needs to be plugged in for x
let's plug in 18
f (18) = [tex]\frac{2}{3} (18) + 3[/tex]
Let's solve
(18 × 2) ÷ 3 = 12
12 + 3 = 15
f (18) = 15
Which of these expressions is equivalent to log(16 • 14)?
CA. 16 log (14)
OB. log(16)+ log(14)
C. log(16) log(14)
D. log(16)-log(14)
Answers
The equivalent expression of log(16 × 14) is log 16 + log 14.
How to find equivalent expression?Two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.
Equivalent expressions are expressions that have similar value or worth but do not look the same.
Therefore, the equivalent expression of log(16 × 14) can be found as follows:
using logarithm law,
The log of a product is the sum of the logs.
logₐ xy = logₐ x + logₐ y
Therefore,
log(16 × 14) = log 16 + log 14
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Find the term t of each continuously compounded account below rounded to the nearest tenth of a year. In the chart, B is the
ending balance, P is the principal, and r is the interest rate expressed as a percent.
Answers
Therefore , the term t of each continuously compounded account is 23.1 years.
What is compound interest ?Compound interest is the practice of adding interest to the principal of a loan or deposit. It happens when interest is reinvested, added to the lent capital rather than paid out, or when the borrower is forced to pay it, resulting in interest being generated on the principal amount plus any accumulated interest the following period. Compound interest is frequently used in economics and finance.
Here,
A) To determine the account's annual growth rate, we must calculate the impact of continual compounding. Similar to the nominal rate (APR) and actual yield (APY) on a bank's interest-bearing account, the yield will be marginally greater due to compounding and indicate actual growth rather than growth rate. Note e is a very lengthy number that can be entered with as many digits as your calculator allows if you don't have a financial calculator handy. e = 2.71828183
A = P[tex]e^{rt}[/tex] But for this, we only need to calculate growth or ert; we don't need to worry about P. When rounding in accordance with the instructions, we set t = 1 for a year and have er (or 2.71828183.03), which equals 1.03045, therefore the account increases by 3.05% annually.
B) 3000 = 1500[tex]e^{0.3t}[/tex] [Divide both sides by 1500]
To isolate t, we must determine the natural logarithm of each side (e's natural logarithm is 1 and 2's natural logarithm is 0.693). 2 = e.03t
0,693 =.03t [Subtract both sides from .03]
23.1 = t
C) Using the formula from section A, first determine the original investment's worth after ten years. A = P A = 1500e.03*10 A = 1500e.3 A = 2,024.79
Next, use the yearly compounding formula to get the value of the second investment.
A = P(1+r)t
A = 1200(1+.045)10
A = 1,863.56
After ten years, the initial investment would be worth more.
Therefore , the term t of each continuously compounded account is 23.1 years.
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In order to add
3/4 and 7/10
you need to find equivalent fractions that have the same denominators. What is a common denominator?
Answers
Thus ,after adding 3/4 and 7/10 we get 29/20 where 20 is common denominator
Describe the common denominator.The smallest common multiple among the denominators of a group of fractions is known as the lowest common denominator (least common denominator) in mathematics. Fraction comparison, addition, and subtraction are made easier.
Here,
In order to add 3/4 and 7/10
we have to find LCM of 4 and 10
which comes out to be = 20
Thus,
addding => 15+`14 /20
=>29/20
Thus ,after adding 3/4 and 7/10 we get 29/20 where 20 is common denominator
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Leia is setting up 417 toy cars. Each car needs 3 batteries to drive.
How many batteries does Leia need for her cars?
Answers
Answer:
Leia needs 3 * 417 = <<3*417=1251>>1251 batteries for her cars.
Step-by-step explanation:
To find out how many batteries Leia needs for her cars, we need to multiply the number of toy cars by the number of batteries needed for each car. Since each car needs 3 batteries and Leia has 417 toy cars, the total number of batteries she needs is 3 * 417 = 1251.
An object has a mass of 10.0 g and a volume of 50.0 mL. What is the density of that object?
5 ml/g
0.2 g/mL
500 g/mL
Answers
Answer:
0.2 g/mL
Step-by-step explanation:
The calculation for density is density=mass/volume.
With a mass of 10g and a volume of 50mL, you divide them to get 0.2
Carla divided 28 bracelets equally among 4 friends. Then, she gave Megan 3 more bracelets. Which equation shows how many bracelets, b, Megan has now?
Answers
The required number of bracelets that in the given question are 10.
What is equation?In math, the meaning of a equation is a numerical explanation that shows that two numerical articulations are equivalent. For example, 3x + 5 = 14 is a condition, where 3x + 5 and 14 are two articulations isolated by an 'equivalent' sign.
According to question:Carla divided 28 bracelets equally among 4 friends.
Number of bracelets each friends has = 28/4 = 7 bracelets
Then Carla gave 3 more bracelets to Magan.
So, Total bracelets Magan has = 7 + 3 = 10 bracelets
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5 - 2x^2+y^3 =-x^3 -3y then find (dy)/(dx) at the point (3,-2)
Answers
The value of the differentiation dy / dx will be 13 / 5.
What is calculus?Calculus, also known as infinitesimal calculus or "the calculus of infinitesimals," is the mathematical study of continuous change, similar to how geometry studies shape and algebra studies generalizations of arithmetic operations.
Given that the expression is 5 - 2x²+y³ =-x³ -3y. The coordinate is (3,-2).
The value of dy / dx is calculated as,
5 - 2x²+y³ =-x³ -3y.
Differentiate the above equation with respect to x,
5 - 2x²+y³ =-x³ -3y
-4x + 3y² dy /dx = 3x² -3dy / dx
-4x - 3x² = (-3y² -3)dy/dx
dy /dx = [-4x - 3x²] / [ (-3y² -3) ]
Substitute the value of x and y in the above equation,
dy /dx = [-4x - 3x²] / [ (-3y² -3) ]
dy /dx = [-4 x 3- 3(3)²] / [ (-3(-2)² -3) ]
dy / dx = [ -12 - 27 ] / [ -15]
dy /dx = -39 /- 15 = 13 / 5
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Find the value of x.
Answers
Answer:
x=142
Step-by-step explanation:
47+95=x
x=142
the two angles equal x
10. The study of the relationships among thermal energy, heat, and work is called dynamics.
True or false
Answers
Answer:
False.
Step-by-step explanation:
False. The study of the relationships among thermal energy, heat, and work is called thermodynamics, not dynamics. Dynamics is a branch of mechanics that deals with the motion of objects and the forces that cause that motion. Thermodynamics, on the other hand, is the study of the relationships between heat, work, and other forms of energy, and how these relationships can be used to understand the behavior of systems. It is a branch of physics that is concerned with the study of heat and its effects on matter.
Write an expression for the height, h, of a cylinder with a volume of 547 cubic inches
in terms of its radius, r.
Answers
The expression for the height (h) in terms of radius r is h = 547/(∏r²).
What is a cylinder?
One of the most fundamental curvilinear geometric shapes, a cylinder has historically been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. In several contemporary fields of geometry and topology, a cylinder can alternatively be characterized as an infinitely curved surface.
Given that the volume of the cylinder is 547 cubic inches.
The volume of a cylinder is V = ∏r²h
Where r is the radius of the cylinder.
h is the height of the cylinder.
Putting V = 547, r = r and h = h in V = ∏r²h
547 = ∏r²h
∏r²h = 547
Divide both sides by ∏r²:
h = 547/∏r².
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Hello, this is a Pythagorean math problem. Help pls.
Answers
The triangles 1, 2, 4, 7 and 8 are right triangles by Pythagorean relationship.
How to determine if a triangle is a right triangle
Triangles are closed figures formed by three line segments, a right triangle is a triangle with one right angle. Right triangles have a long side known as hypotenuse (r) and two shorter sides known as legs (x, y), whose Pythagorean relationship is shown below:
r = √(x² + y²)
Now we proceed to check whether each triangle is a right triangle or not:
Case 1:
D = √(8² + 6²)
D = 10 (YES)
Case 2:
D = √(5² + 12²)
D = 13 (YES)
Case 3:
D = √(12² + 9²)
D = 15 (NO)
Case 4:
D = √(8² + 15²)
D = 17 (YES)
Case 5:
D = √(4² + 9²)
D ≈ 9.849 (NO)
Case 6:
D = √(14² + 5²)
D ≈ 14.866 (NO)
Case 7:
D = √(20² + 15²)
D = 25 (YES)
Case 8:
D = √(24² + 10²)
D = 26 (YES)
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If x + 3y^(1/3) = y what is dy/dx at the point (2,8)
Answers
The dy/dx at point (2,8) is 4/3
How to find dy/dx at a point?From the information illustrated, x + 3y^(1/3) = y and point (2,8)
Find the derivative of this equation with respect to x and solve for dy/dx:
x + 3y^(1/3) = y
1 + [3 × 1/3 × y^(-2/3)]dy/dx = dy/dx
1 + [y^(-2/3)]dy/dx = dy/dx
dy/dx - [y^(-2/3)]dy/dx = 1
dy/dx [1- y^(-2/3)] = 1
dy/dx = 1/ [1 - y^(-2/3)]
At point (2,8), x = 2 and y = 8.
Substitute into dy/dx:
dy/dx = 1/ [1 - y^(-2/3)]
dy/dx = 1/ [1 - 8^(-2/3)]
dy/dx = 1/ [1 - 1/4)]
dy/dx = 4/3
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II. Given that cos A = 0.42 and sin B = 0.73, evaluate
i. sin (A – B)
ii. cos (A – B)
iii. tan (A + B)
Answers
The trigonometric identities values are;
i. sin (A – B) = -0.15
ii. cos (A – B) = 0.5368
iii. tan (A + B) = -3.536
How to solve trigonometric Identity?
We are given;
cos A = 0.42
sin B = 0.73
Since cos A = 0.42 , then sin A = 0.58
Since sin B = 0.73, then cos B = 0.27
i) sin (A – B)
By trigonometric identities, we know that;
sin (A – B) = sinA cos B - cosA sinB
Thus;
sin (A – B) = (0.58 * 0.27) - (0.42 * 0.73)
sin (A – B) = -0.15
ii) cos (A – B)
By trigonometric identities, we know that;
cos (A – B) = cosA cosB + sinA sinB
cos (A – B) = (0.42 * 0.27) + (0.58 * 0.73)
cos (A – B) = 0.5368
iii) tan (A + B)
By trigonometric identities, we know that;
(tan A + tan B)/(1 - tanAtanB)
tan A = sinA/cosA = 0.58/0.42 = 1.389
tan B = sinB/cosB = 0.73/0.58 = 1.259
Thus;
tan (A + B) = (1.389 + 1.259)/(1 - (1.389 * 1.259))
tan (A + B) = -3.536
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