Answer:
building: 222.61 ftflagpole: 32.15 ftStep-by-step explanation:
The tangent function of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, the adjacent side is 500 ft, so we have ...
tan(24°) = (building height)/(500 ft)
building height = (500 ft)tan(24°) ≈ 222.61 ft
__
Similarly, the height to the top of the flagpole is ...
total height = (500 ft)tan(27°) ≈ 254.76 ft
The length of the flagpole is the difference of these heights:
flagpole length = 254.76 ft -222.61 ft = 32.15 ft
The height of the building is 222.61 ft; the length of the flagpole is 32.15 ft.
The height of the building is approximately 250.4 feet, and the length of the flagpole is approximately 24.4 feet, according to the principle of trigonometry using tangent function.
Explanation:The question deals with a mathematical concept called trigonometry. Specifically, we will use the tangent function. Tangent of an angle in a right triangle is the ratio of the side opposite to the angle over the side adjacent to the angle.
Let's denote H as the height of the building and F as the length of the flagpole. We know that Tan(24°) = H / 500 and Tan(27°) = (H+F) / 500. Now we can solve these two equations to find H and F.
First, calculate H. Using the first equation, H = Tan(24°) * 500 ≈ 250.37 feet.
Next, calculate H+F. Using the second equation, H+F = Tan(27°) * 500 ≈ 274.73 feet.
Then, calculate F by subtracting H from the second result. So F = 274.73 - 250.37 = 24.36 feet.
Therefore, the height of the building is approximately 250.4 feet, and the length of the flagpole is approximately 24.4 feet.
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Suppose a and b are the solutions to the quadratic equation 2x^2-3x-6=0. Find the value of (a+2)(b+2).
Answer:
(a+2)(b+2) = 4
Step-by-step explanation:
We are given the following quadratic equation:
[tex]2x^2-3x-6=0[/tex]
Let a a and b be the solution of the given quadratic equation.
Solving the equation:
[tex]2x^2-3x-6=0\\\text{Using the quadratic formula}\\\\x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\\\\\text{Comparing the equation to }ax^2 + bx + c = 0\\\text{We have}\\a = 2\\b = -3\\c = -6\\x = \dfrac{3\pm \sqrt{9-4(2)(-6)}}{4} = \dfrac{3\pm \sqrt{57}}{4}\\\\a = \dfrac{3+\sqrt{57}}{4}, b = \dfrac{3-\sqrt{57}}{4}[/tex]
We have to find the value of (a+2)(b+2).
Putting the values:
[tex](a+2)(b+2)\\\\=\bigg(\dfrac{3+\sqrt{57}}{4}+2\bigg)\bigg(\dfrac{3-\sqrt{57}}{4}+2\bigg)\\\\=\bigg(\dfrac{11+\sqrt{57}}{4}\bigg)\bigg(\dfrac{11-\sqrt{57}}{4}\bigg)\\\\=\dfrac{121-57}{156} = \dfrac{64}{16} = 4[/tex]
Find all x that satisfy the inequality (2x 10)(x 3)<(3x 9)(x 8). Express your answer in interval notation.
To find all x that satisfy the inequality (2x + 10)(x + 3) < (3x + 9)(x + 8), we expand and simplify the inequality, solve for x by considering the sign of each factor, and express the answer in interval notation. The solution is (-7, -6).
Explanation:To find all x that satisfy the inequality (2x + 10)(x + 3) < (3x + 9)(x + 8), we will first expand and simplify the inequality. Then, we will solve for x by considering the sign of each factor.
Start by expanding both sides of the inequality:The interval notation for the solution is (-7, -6). This means that all values of x between -7 and -6 (exclusive) satisfy the inequality.
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At the movie theater child admission is 6000 and $.30 an adult mission is $9.60 on Wednesday 172 tickets were sold for a total sales of 1403 70 how many tickets were sold that day
Answer:
The number of child tickets sold was 75 and the number of adult tickets sold was 97
Step-by-step explanation:
The correct question is
At the movie theater child admission is $6.30 an adult admission is $9.60 on Wednesday 172 tickets were sold for a total sales of $1403.70 How many child tickets and adult tickets were sold that day?
Let
x ----> number of child tickets sold
y ----> number of adult tickets sold
we know that
[tex]x+y=172[/tex] ----> equation A
[tex]6.30x+9.60y=1,403.70[/tex] -----> equation B
Solve the system by graphing
Remember that the solution of the system is the intersection point both lines
Using a graphing tool
The intersection point is (75,97)
see the attached figure
therefore
The number of child tickets sold was 75 and the number of adult tickets sold was 97
To solve this problem, set up a system of equations based on the given information. Then use the substitution method to solve for the variables. 69 child tickets and 103 adult tickets were sold on that day.
Explanation:To solve this problem, we can set up a system of equations.
Let's define the number of child tickets sold as x and the number of adult tickets sold as y.
We know that the cost of a child ticket is $6.00 and the cost of an adult ticket is $9.60.
So, the total cost of all the child tickets is 6x and the total cost of all the adult tickets is 9.60y.
Given that a total of 172 tickets were sold for a total of $1403.70, we can set up the following equations:
6x + 9.60y = 1403.70 (equation 1)x + y = 172 (equation 2)Now we can solve this system of equations using any method we prefer. Let's use the substitution method.
Rearrange equation 2 to solve for x: x = 172 - ySubstitute this expression for x into equation 1: 6(172 - y) + 9.60y = 1403.70Simplify and solve for y: 1032 - 6y + 9.60y = 1403.70Combine like terms: 3.60y = 371.70Divide both sides by 3.60: y = 103Substitute this value for y back into equation 2 to find x: x + 103 = 172Subtract 103 from both sides: x = 69Therefore, 69 child tickets and 103 adult tickets were sold on that day.
A man buys 3 burgers and 2 jumbo deluxe fries for $7.40. A woman buys one burger and 4 jumbo deluxe fries for $7.80. How much is the burger and how much are the fries?Select one:a. Burger = $1.40, Fries = $1.60b. Burger = $1.50, Fries = $1.80c. Burger = $2.80, Fries = $2.00d. Burger = $2.00, Fries = $2.80
Answer: A. See photo for work.
Step-by-step explanation:
Answer:the cost of one burger is $1.4 and the cost of one fry is $1.6
Step-by-step explanation:
Let x represent the cost of one burger.
Let y represent the cost of one fry.
A man buys 3 burgers and 2 jumbo deluxe fries for $7.40. This means that
3x + 2y = 7.4 - - - - - - - - - 1
A woman buys one burger and 4 jumbo deluxe fries for $7.80. It means that
x + 4y = 7.8 - - - - - - - - -2
Multiplying equation 1 by 1 and equation 2 by 3, it becomes
3x + 2y = 7.4
3x + 12y = 23.4
Subtracting
- 10y = - 16
y = - 16/- 10 = 1.6
Substituting y = 1.6 into equation 2, it becomes
x + 4 × 1.6 = 7.8
x = 7.8 - 6.4 = 1.4
Remy, from the blog "Rapping with Remy," asked his readers, "If you had the chance, would you want to be famous?" Of the 2,200 readers who responded, 42% said no, and all were female. What does this show?
possible answers:
No meaningful conclusion is possible without knowing something more about the characteristics of Remy's readers.
The survey is meaningless because of undercoverage bias.
The survey would have been more meaningful if Remy had picked a random sample of the readers who responded.
the survey would have been more meaningful if Remy had used a control group.
This was a legitimate sample, drawn randomly from his readers, and is of adequate size to allow the conclusion that most of Remy's readers want to be famous.
Understanding the representativeness of samples in surveys is crucial for drawing meaningful conclusions. In these examples, the lack of information about the characteristics of Remy's readers and the undercoverage bias in the radio station survey make the conclusions unreliable.
Explanation:The question is about the representativeness of samples in surveys. In the case of Remy's blog survey, the fact that only female readers responded and that more information about their characteristics is needed suggests that no meaningful conclusion is possible without knowing something more about the characteristics of Remy's readers.
In the case of the local radio station survey, convenience sampling was used by only surveying people attending the concert events, which may not accurately represent the entire 20,000 listener population. Therefore, the survey is meaningless because of undercoverage bias.
A wildlife management organization is interested in estimating the number of moose in a particular region. Organization employees divide the region into 10 sections and randomly select four sections to survey the number of moose present. What sampling method is being used? a. Cluster sampling b. Systematic sampling c. Judgment sampling d. Stratified random sampling
Answer: Why are they increasing the moose?
Step-by-step explanation: Well I’m assuming this is in Canada. therefore, this has to be D
A music store is offering a Frequent Buyers Club membership. The membership costs $22 per year, and then a member can buy CDs at a reduced price. If a member buys 17 CDs in one year, the cost is $111.25. Determine the cost of each CD for a member.
Answer:the cost of each CD for a member is $5.25
Step-by-step explanation:
The membership costs $22 per year, and then a member can buy CDs at a reduced price.
Let x represent the cost of each CD for a member.
Let n represent the number of CDs that each member buys. This means that the total cost of buying n CDs for a member for a year would be
22 + nx
If a member buys 17 CDs in one year, the cost is $111.25. This means that
17x + 22 = 111.25
17x = 111.25 - 22 = 89.25
x = 89.25/175 = 5.25
one college states that the number of men, M, and the number of women, W, receiving bachelor degrees t years since 1980 can be modeled by the function M(t)=526-t and W(t)=474+2t, respectively. Let N be the total number of students receiving bachelor's degrees at the college t years since 1980. Write an expression for N(t)
Answer:
[tex]N(t) = 1000 +t[/tex]
Step-by-step explanation:
It is given that,
Number of Men receiving degree are = M(t) = 526-t
Number of women receiving degree are = W(t) = 74+2t
The total number of students receiving the degree is given by,
sum of both Men and Women receiving the degrees.
Thus, N(t) = M(t) + W(t)
N(t) = (526-t) + (474+2t)
N(t) = 1000 + t
Thus, the number of students receiving the degree is given by N(t) = 1000 + t .
Answer:
Step-by-step explanation:
Some kangaroos can cover 30 feet in a single jump if a kangaroo counts jump like that 150 times in a row how much farther wound it need to go to cover a mile
Answer:
Distance farther the Kangaroos need to go to cover a mile = 780 feet
Step-by-step explanation:
Given:
Kangaroos cover 30 feet in a single jump.
Kangaroos can make such jumps 150 times in a row.
To find the distance farther it needs to go to cover a mile.
Solution:
Using unitary method to find distance covered by kangaroos in a row.
If in a single jump Kangaroos cover = 30 feet
So, in 150 jumps it will cover = [tex]30\ ft \times 150[/tex] = 4500 feet
We know 1 mile = 5280 feet
Thus, distance farther the Kangaroo needs to go to cover a mile will be = [tex]5280\ ft-4500\ ft[/tex] = 780 feet
A las 9 de la mañana,la temperatura en Valcorto era de -8 grados . A las 12 horas era dos grados mayor ,a las 15 horas tres grados mas que a las 12 ,y a las 21 horas nueve grados menos que a las 15 ¿que temperatura habia cada hora citada?
Answer:
[tex]T_{9hr}=-8 C[/tex]
[tex]T_{12hr}=-6 C[/tex]
[tex]T_{15hr}=-3 C[/tex]
[tex]T_{21hr}=-12 C[/tex]
Step-by-step explanation:
The initial temperature at 9 am:
[tex]T_{9hr}=-8 C[/tex]
At 12 hr is 2 degrees higher:
[tex]T_{12hr}=T_{9hr}+2 C=-6 C[/tex]
At 15 hr is 3 degrees higher than 12hr:
[tex]T_{15hr}=T_{12hr}+3 C=-3 C[/tex]
Finally, at 21 hr, is 9 degrees lower than 15 hr:
[tex]T_{21hr}=T_{15hr}-9 C=-12 C[/tex]
Find all solutions to the equation in the interval [0, 2π).
cos x = sin 2x
pi divided by two., three pi divided by two.
pi divided by six., pi divided by two., five pi divided by six., three pi divided by two.
0, π
0, pi divided by six, five pi divided by six., π
Answer:
x = π/6, π/2, 5π/6, 3π/2
Step-by-step explanation:
cos x = sin(2x)
Use double angle formula.
cos x = 2 sin x cos x
Move everything to one side and factor.
cos x − 2 sin x cos x = 0
cos x (1 − 2 sin x) = 0
Set each factor to 0 and solve.
cos x = 0
x = π/2, 3π/2
1 − 2 sin x = 0
sin x = 1/2
x = π/6, 5π/6
The total solution is:
x = π/6, π/2, 5π/6, 3π/2
Final answer:
The solutions to the equation cos x = sin 2x in the interval [0, 2π) are π/6, π/2, 5π/6, and 3π/2, derived by using the identity sin 2x = 2 sin x cos x and considering cases for cos x = 0.
Explanation:
To find all solutions to the equation cos x = sin 2x in the interval [0, 2π), we first need to use a trigonometric identity to express both sides of the equation with either sine or cosine. The identity sin 2x = 2 sin x cos x can be used here. Substituting it into our original equation, we get:
cos x = 2 sin x cos x
To solve this equation, we can divide both sides by cos x, given that cos x ≠ 0:
1 = 2 sin x
sin x = 1/2
Using the unit circle or trigonometric tables, we know that sin x takes the value of 1/2 at x = π/6 and x = 5π/6 in the interval [0, 2π). Additionally, we must consider the case when cos x = 0 to avoid division by zero. This occurs at x = π/2 and x = 3π/2, which are also solutions to the original equation given that sin(2(π/2)) = sin(π) = 0 and sin(2(3π/2)) = sin(3π) = 0, which are equal to cos(π/2) and cos(3π/2) respectively. Thus, the complete set of solutions in the interval [0, 2π) is π/6, 5π/6, π/2, and 3π/2.
The crude association between occupational exposure and lung cancer (outcome) among the study sample was 8.4. When data were stratified by alcohol intake, the associations were 2.2 (for non-drinkers) and 14.5 (for drinkers). Assume that all (crude, and stratified) measures of association were statistically significant. Alcohol use was (use 15% difference rule):
a. An effect modifier
b. A potential confounder
c. A random error
d. A confounder
Answer:
Option D
Step-by-step explanation:
This are variables that have relationship with the outcome and exposure that is the confounding effect of alcohol on cancer is the fact that an alcohol consumers are more likely to smoke and smokers are likely to have cancer.
The average score of 100 teenage boys playing a computer game was 80,000 with a population standard deviation of 20,000. What is the 95% confidence interval for the true mean score of all teenage boys?
The 95% confidence interval for the true mean score of all teenage boys is approximately 76,080 to 83,920.
To calculate the 95% confidence interval for the true mean score of all teenage boys, we'll use the formula for the confidence interval:
[tex]\[ \text{Confidence Interval} = \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean,
- Z is the Z-score corresponding to the desired confidence level,
- [tex]\(\sigma\)[/tex] is the population standard deviation, and
- n is the sample size.
For a 95% confidence interval, the Z-score is approximately 1.96.
Given:
- Sample mean [tex](\(\bar{x}\))[/tex] = 80,000
- Population standard deviation (\(\sigma\)) = 20,000
- Sample size (n) = 100
[tex]\[ \text{Confidence Interval} = 80,000 \pm 1.96 \left( \frac{20,000}{\sqrt{100}} \right) \][/tex]
Calculate the standard error (SE):
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20,000}{\sqrt{100}} = 2,000 \][/tex]
Now substitute the values into the formula:
[tex]\[ \text{Confidence Interval} = 80,000 \pm 1.96 \times 2,000 \][/tex]
[tex]\[ \text{Confidence Interval} = 80,000 \pm 3,920 \][/tex]
The 95% confidence interval is from 80,000 - 3,920 to 80,000 + 3,920.
On Monday morning, Susan bought 2.5 pounds of grapes at the supermarket. That day, Susan ate some grapes and had 1.75 pounds of grapes left. What is the percent decrease of pounds of grapes on Monday?
Answer:
Step-by-step explanation:
On Monday morning, Susan bought 2.5 pounds of grapes at the supermarket. That day, Susan ate some grapes and had 1.75 pounds of grapes left. This means that the amount of grapes that she ate would be 2.5 - 1.75 = 0.75
The percentage decrease of pounds of grapes on Monday would be the amount that she ate on Monday divided by the initial amount times 100. It becomes
0.75/2.5 × 100 = 30℅
What is the equation of the line with m = 6 that goes through the point (1, 4)?
A. y – 4 = 6(x + 1)
B. y – 4 = 6(x – 1)
C. y + 4 = 6(x – 1)
D. y + 4 = 6(x + 1)
Since it's a line we are talking about linear equation of a form
[tex]f(x)=mx+n[/tex]
where [tex]m[/tex] is slope and [tex]n[/tex] is y-intercept.
Our particular line has a form of
[tex]f(x)=6x+n[/tex]
So we are missing the y-intercept.
To find y-intercept [tex]n[/tex] we insert the coordinates of point [tex]P(x,f(x))\to P(1,4)[/tex] and solve for [tex]n[/tex]
[tex]
4=6\cdot1+n \\
n=-2
[/tex]
So the final form of the line is
[tex]y=6x-2[/tex]
Or as offered in the answers
[tex]y-4=6(x-1)[/tex]
The answer is B.
Hope this helps.
The greatest common divisor of two positive integers less than $100$ is equal to $3$. Their least common multiple is twelve times one of the integers. What is the largest possible sum of the two integers?
Answer:
129
Step-by-step explanation:
Let a and b be two numbers.
We have been given that the greatest common divisor of two positive integers less than 100 is equal to 3. We can represent this information as [tex]GCD(a,b)=3[/tex].
Their least common multiple is twelve times one of the integers. We can represent this information as [tex]LCM(a,b)=12a[/tex].
Now, we will use property [tex]GCD(x,y)*LCM(x,y)=xy[/tex].
Upon substituting our given values, we will get:
[tex]3*12a=ab[/tex]
[tex]36a=ab[/tex]
Switch sides:
[tex]ab=36a[/tex]
[tex]\frac{ab}{a}=\frac{36a}{a}[/tex]
[tex]b=36[/tex]
Now, we need to find a number less than 100, which is co-prime with 12 after dividing by 3.
The greatest multiple of 3 less than 100 is 99, but it is not co-prime with 12 after dividing by 3.
Similarly 96 is also not co-prime with 12 after dividing by 3.
We know that greatest multiple of 3 (less than 100), which is co-prime with 12, is 93.
Let us add 36 and 93 to find the largest possible sum of the required two integers as:
[tex]36+93=129[/tex]
Therefore, the required largest possible sum of the two integers is 129.
If a lineman can install 12 insulators in 183/4 hours, how many insulators should he be able to install in 281/8 hours?
Answer: 9
Step-by-step explanation:
12 insulators = 183/4 hours
x = 281/8
x × 183/4 = 12 ×281/ 8
183x/4 = 6 × 281/4
183x / 4 = 1686 / 4
4 cancel out 4
183x = 1686
Divide bothside by 183
x = 1686/183
x= 9.21311
x = 9 approximately.
Answer:
$0.04 per foot
Step-by-step explanation:
24. You use math in day-to-day routines when grocery shopping, going to the bank or mall, and while cooking. How do you imagine you will use math in your healthcare career?
Answer: in dispensing medications, in the use of measurement equipments and formulas for converting units, to obtain reliable data of patients overtime to be able to predict, and diagnose patients easily
Step-by-step explanation:
Maths is used in health care in the dispensing of medications they translate medical order into dosages, calculate dosage based on body weight, determine the amount of drugs to be dispensed. Maths also helps in the conversion of units e.g milligram to gram, Fahrenheit to degree Celsius etc.
It also help in the measurement of body temperature, pulse rate, breathing rate etc.
Among others uses, maths is also used to generate reliable data of patients overtime and help in quick diagnosis and treatment of patients.
To determine whether there is a relationship between the type of school attended and verbal reasoning scores for Irish students, three samples with 25 students, in each group, were randomly selected from data used by Raferty and Hout (1985). One group of students attended secondary school, the second group of students attended vocational school, and the third group consisted of students who attended only primary school.
Here are the three sample standard deviations for the verbal reasoning scores for the three groups (secondary school, vocational school, and primary school only):
Based on this information, do the data meet the condition of equal population standard deviations for the use of the ANOVA?
A. Yes, because 14.18 − 11.71 < 2.
B. Yes, because 14.18/11.71 < 2.
C. No, because the standard deviations are not equal.
Answer:
B. Yes, because 14.18/11.71 < 2.
Step-by-step explanation:
When the ratio of the largest sample standard deviation to the smallest sample standard deviation is less than 2, the condition is said to be met.
I WILL GIVE BRAINLIEST FOR THE CORRECT ANSWER AND POINTS!
Answer:
36.5
Step-by-step explanation:
s² = [∑(min value - median value)² + (max value - median value)²] / (N - 1)
s² = [(23 - 59.5)² + (96 - 59.5)²] / (3 - 1)
s² = 2664.5 / 2
s² = 1332.25
s = √1332.25
s = 36.5
What is the value of cos-1 (-√3/2
The value of cos-1 (-√3/2) represents the angle that has the cosine of -√3/2, and it is found in the second or third quadrant. The angles corresponding to these values in the unit circle are 120° or 240°.
Explanation:The value of cos-1 (-√3/2) refers to the angle whose cosine is -√3/2. This angle would be found in the second or third quadrant since cosines are negative in these quadrants. For the value of -√3/2, it corresponds to the angle 120° or 240° in the unit circle. Therefore, the value of cos-1 (-√3/2) is 120° or 240°.
Remember, inverse trigonometric functions deal with the range and the quadrant in which an angle lays. While working with inverse trigonometric functions, pay attention to the unit circle, and to the fact that cosine is negative in the second and third quadrants.
The value of cos-1 (-√3/2) is 150 degrees or 5π/6 radians.
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KarlaPurchase a swimsuit on Amazon the original price of the coat was 75.50 she use the coupon code to receive a 25% discount the website applied a 10% service fee for the discount price Karla swimsuit Was less then original price by what percent
Answer:
Step-by-step explanation:
The initial price of the swimsuit that Karla purchased at Amazon was 75.5
she use the coupon code to receive a 25% discount. This means that the value if the discount would be
25/100 × 75.5 = 0.25× 75.5 = 18.86
Therefore, the discounted price of the swimsuit is 75.5 - 18.875 = 56.625
The website applied a 10% service fee for the discount price. This means that the value of the service fee would be
10/100 × 56.625 = 0.1 × 56.625 = 5.6625.
The amount that Karla pays for the swimsuit would be
56.625 + 5.6625 = 62.2875
The difference between the original price and the price that Karla paid would be 75.5 - 62.2875 = 13.2125
The percentage by which the amount that Karla paid was lesser than the original price would be
13.2125/ 75.5 × 100 = 17.5%
Is the binomial a factor of the polynomial function?
f(x)=x^3+4x^2−25x−100
(I'm not sure if the highlighted answers are correct, help!!!)
Answer:
YES
NO
NO
Step-by-step explanation:
The given polynomial is: [tex]$ f(x) = x^3 + 4x^2 - 25x - 100 $[/tex]
(x - a) is a factor of a polynomial iff x = a is a solution to the polynomial.
To check if (x - 5) is a factor of the polynomial f(x), we substitute x = 5 and check if it satisfies the equation.
∴ f(5) = 5³ + 4(5)² - 25(5) - 100
= 125 + 100 - 125 - 100
= 225 - 225
= 0
We see, x = 5 satisfies f(x). So, (x - 5) is a factor to the polynomial.
Now, to check (x + 2) is a factor.
i.e., to check x = - 2 satisfies f(x) or not.
f(-2) = (-2)³ + 4(-2)² - 25(-2) - 100
= -8 + 16 + 50 - 100
= -108 + 66
≠ 0
Therefore, (x + 2) is not a factor of f(x).
To check (x - 4) is a factor.
∴ f(4) = 4³ + 4(4)² - 25(4) - 100
= 64 + 64 - 100 - 100
= 128 - 200
≠ 0
Therefore, (x - 4) is not a factor of f(x).
Answer:
The answers are (x-5) YES (x+2) NO (x-4) NO
Step-by-step explanation:
I took the test :)
Abc is an isosceles triangle with ba=bc d lies on ac.Abd is an isosceles triangle with ab=ad angle abd=72 show that triangle bcd is isosceles.You must give a reason for each working out.
Answer:
ADB=72˚. base angles in an isosceles triangle are equal
72+72=144 180-144=36 BAD=36 ˚Angles in a triangle =180˚
180-72=108˚ BDC=108˚ angles on a straight line=180˚
Step-by-step explanation:
only three marks for this one
By exploiting the properties of isosceles triangles and the sum of angles in a triangle, we deduce that triangle BCD is isosceles with BC = BD.
Explanation:The student is asking how to prove that triangle BCD is isosceles given that triangles ABC and ABD are also isosceles with AB = BC and AB = AD, respectively, and angle ABD = 72 degrees. We can begin by noting that the sum of the angles in any triangle is 180 degrees. Because triangle ABD is isosceles with AB = AD, the angles ABD and ADB are equal, and since angle ABD is 72 degrees, angle ADB is also 72 degrees. Therefore, angle BAD, the remaining angle in triangle ABD, must be 180 - 72 - 72 = 36 degrees.
Since triangle ABC is also isosceles with AB = BC, angles ABC and BAC are equal. Angle BAD is part of angle BAC, which means angle BAC is also 36 degrees (since they both include angle BAD). Consequently, angle ABC is 36 degrees. The base angles of an isosceles triangle are equal, so angle BCA must also be 36 degrees. Because ABC is isosceles with AB = BC, and we have determined that angles ABC and BCA are equal, we conclude that angles BCD and CBD must also be equal, making triangle BCD isosceles, with BC = BD.
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Stacey owns a lot that has 180 feet of front footage and contains 36,000 square feet. She purchases two lots adjacent to each side of his lot. These side lots are each 200 feet deep and contain 19,000 square feet. What is the total front footage of all three lots?
Step-by-step explanation:
Front footage of first lot = 180 ft
Area of each side lot = 19000 ft²
Depth of each side lot = 200 ft
We have
Area = Depth x Front footage
19000 = 200 x Front footage
Front footage = 95 ft
Total front footage = Front footage of first lot + 2 x Front footage of each side lot
Total front footage = 180 + 2 x 95
Total front footage = 180 + 190
Total front footage = 370 ft
The total front footage of all three lots is 370 ft
Four milliliters of ink A is added to 10 milliliters of ink B, forming a mixture of inks. Ink A contains 10% blue pigment. Ink B contains an unknown percentage of blue pigment. The mixture of the two inks contains 35% blue pigment. What percentage of blue pigment is in ink B?
A. 45%
B. 49%
C. 60%
D. 75%
Answer:
The answer to your question is letter A
Step-by-step explanation:
Percent of blue pigment Milliliters
ink A 10% 4
ink B --- 10
Total volume 35% 14
Formula
Total percent x total volume = (percent ink A x A milliliters) +
(percent ink B x B milliliters)
Substitution
(0.35 x 14) = ( 0.1 x 4) + (B x 10)
Solve for B
4.9 = 0.4 + 10B
10B = 4.9 - 0.4
10B = 4.5
B = 4.5/10
B = 0.45 x 100
B = 45%
Represent each of these relations on {1, 2, 3, 4} with a matrix (with the elements of this set listed in increasing order). a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)} c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)} d) {(2, 4), (3, 1), (3, 2), (3, 4)}
The concept in the question is about representing relations as matrices. In the matrices, the rows and columns correspond to the numbers in the set {1, 2, 3, 4}. The positions in the matrices that are filled with 1s correspond to the numbers in the given relationships.
Explanation:The relations can be represented on the set {1, 2, 3, 4} with a matrix as follows:
a) For this relationship, a 4x4 matrix can be created. The rows and columns correspond to the numbers in the set {1, 2, 3, 4}. The elements of the set [tex]{(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}[/tex] correspond to positions in the matrix that will be filled with 1s, while the rest will be filled with 0s.
b) A matrix for this relationship would look similar to the above matrix but with different elements in the set [tex]{(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)}[/tex]. In this case, the position of 1s and 0s in the matrix changes.
c) The matrix corresponding to this relationship would have 1s in positions corresponding to the elements of the set [tex]{(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}.[/tex]
d) The last matrix would have 1s only at the positions corresponding to the relationship {(2, 4), (3, 1), (3, 2), (3, 4)}.
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Pete wants to make turkey sandwiches for tow friends and himself.He wants each sandwich to contain 3.5 ounces of turkey.How many ounces of turkey does he need
Pete needs 10.5 ounces of turkey.
Step-by-step explanation:
Given,
Number of friends = 2
One for Pete.
Total sandwiches to be made = 2+1 = 3
Quantity of turkey in one sandwiches = 3.5 ounces
For finding the quantity of turkey for three sandwiches, we will multiply.
Turkey for 3 sandwiches = 3*3.5 = 10.5 ounces
Pete needs 10.5 ounces of turkey.
Keywords: multiplication, addition
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Solve for x. The triangles in each pair are similar.
Answer:
Step-by-step explanation:
You need to pay very close attention to the triangle similarity statement. This says that triangle NML is similar to triangle NVU. But if you look at the way that triangle NVU is oriented in its appearance, it's laying on its side. We need to set it upright so that angle N is the vertex angle, angle V is the base angle on the left, and angle U is the base angle on the right. When we do that we see that sides NV and NM are corresponding and exist in a ratio to one another; likewise with sides VU and ML. Setting up the proportion:
[tex]\frac{NV}{NM}=\frac{VU}{ML}[/tex]
Filling in:
[tex]\frac{12}{36} =\frac{9}{9x}[/tex]
Cross multiply to get
324 = 108x
and x = 3
ABCD is a parallelogram AC = 15, m∠ BAC = 22° , m∠DAC = 27° Find: AB and BC
Answer:
[tex]AB=9.02\ units[/tex]
[tex]BC=7.45\ units[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
Remember that in a parallelogram opposites sides are parallel and congruent, opposites angles are congruent and consecutive angles are supplementary
step 1
Find the measure of angle ACB
we have
[tex]m\angle BAC=22^o[/tex] ----> given problem
[tex]m\angle ACB=m\angle DAC[/tex] ----> by alternate interior angles
[tex]m\angle DAC=27^o[/tex] ----> given problem
so
[tex]m\angle ACB=27^o[/tex]
step 2
Find the measure of angle ABC
The sum of the interior angles in any triangle must be equal to 180 degrees
In the triangle ABC of the figure
[tex]m\angle BAC+m\angle ACB+m\angle ABC=180^o[/tex]
substitute the given values
[tex]22^o+27^o+m\angle ABC=180^o[/tex]
[tex]49^o+m\angle ABC=180^o[/tex]
[tex]m\angle ABC=180^o-49^o[/tex]
[tex]m\angle ABC=131^o[/tex]
step 3
Find the length side AB
In the triangle ABC
Applying the law of sines
[tex]\frac{AC}{sin(ABC)}=\frac{AB}{sin(ACB)}[/tex]
substitute the given values
[tex]\frac{15}{sin(131^o)}=\frac{AB}{sin(27^o)}[/tex]
[tex]AB=\frac{15}{sin(131^o)}(sin(27^o))[/tex]
[tex]AB=9.02\ units[/tex]
step 4
Find the length side BC
In the triangle ABC
Applying the law of sines
[tex]\frac{AC}{sin(ABC)}=\frac{BC}{sin(BAC)}[/tex]
substitute the given values
[tex]\frac{15}{sin(131^o)}=\frac{BC}{sin(22^o)}[/tex]
[tex]BC=\frac{15}{sin(131^o)}(sin(22^o))[/tex]
[tex]BC=7.45\ units[/tex]
To find the lengths of AB and BC in the parallelogram ABCD, one can use the sine and cosine trigonometric ratios along with the given lengths and angles of the parallelogram. AB is calculated using the sine of angle DAC and the length of AC, while BC is equal to AD, which can be found using the cosine of angle DAC and the length of AC.
Explanation:The student has a parallelogram ABCD with given measures and needs to find the lengths of sides AB and BC. To find side AB, we can use trigonometric ratios in the triangle ACD, as angle DAC is known and AC is the hypotenuse. For side BC, we could use the fact that opposite sides in a parallelogram are equal, thus BC will equal AD which is the adjacent side to angle DAC in triangle ACD.
For AB, which is the opposite side to angle DAC, we can use:
AB = AC × sin(angle DAC) = 15 × sin(27°).
The specific value can be computed with a calculator.
Since parallelogram ABCD has opposite sides equal:
BC = AD
Therefore, we can find AD using the trigonometric ratio involving the cosine of angle DAC:
AD = AC × cos(angle DAC) = 15 × cos(27°).
The computed value will be the length of side BC.