solve tan(2θ)+tan(θ)=0 exactly
Complete the square to put he equation in convenient form for graphing. Then graph the conic section
16x^2+9y^2+96x-18y+9=0
The sum of nine and two times a number is 35. What is the number?
A rectangular room is 1.5 times as long as it is wide, and its perimeter is 35 meters. find the dimesions of the room
The width of the room is 7 meters and the length is 10.5 meters.
Explanation:To find the dimensions of the rectangular room, let's assume the width of the room is x meters. According to the given information, the length of the room is 1.5 times the width, so the length would be 1.5x meters.
The perimeter of a rectangle is calculated by adding twice the length to twice the width. So, we have the equation:
2x + 2(1.5x) = 35
Simplifying the equation, we get:
2x + 3x = 35
Combining like terms, we have:
5x = 35
Dividing both sides of the equation by 5, we find that x = 7.
Therefore, the width of the room is 7 meters and the length is 1.5 times the width, which is 1.5 * 7 = 10.5 meters.
Together, Louisa and Jill scored more than 30 points in the basketball game. If Jill scored 12 points, how many points P did Louisa score? A. P > 18
Answer:
Option A) P > 18 is the correct option.
Step-by-step explanation:
Louisa and Jill scored jointly in basketball = Score of Louisa + score of Jill
Joint score is greater than 30 points
So Score of Louisa P + score of Jill > 30
If Jill scored 12 points then inequality becomes as
score of Louisa P + 12 > 30
Now by subtracting 12 from both side
score of Louisa P + 12 - 12 > 30 - 12
score of Louisa P > 18
The area of a rectangle wall of a barn is 234 square feet. It's length is 8 feet longer than twice its width. Find the length and width of the wall of the barn.
The length of the wall is : 26 ft.
and width of the wall is: 9 ft.
What is area of a rectangle?
The shape/polygon of a rectangle is two dimensional, having four sides, four vertices, and four right angles. The rectangle's two opposing sides are equal and parallel to one another. The space a rectangle occupies is known as its area. The area of a rectangle can also be defined as the region inside its border.
The area of a rectangle is: Area= length × width
Solution:
area = 234
length = ?
width = ?
Given, length is 8 ft greater than twice the width.
Now, l = 2w ₊ 8
l × w = 234 (given)
Substitute for l
(2w ₊ 8)w = 234
multiply
2w² ₊ 8w = 234
2w² ₊ 8w ₋ 234 = 0
Now divide by 2 to eliminate the coefficients of w.
w² ₊ 4w ₋ 117=0
get the factors for the above equation.
We get the factors as: 9 and ₋13,( ₋13 is not applicable as width cannot be negative)
Hence we get the width as 9, now substitute it in:
l = 2w ₊ 8
l = 2(9) ₊ 8
l = 26 ft,
Hence the length and width of the rectangle is 26 ft and 9 ft respectively.
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what is the answer of 21 + m = 33
Using the digits 3, 4, 5, 6, 7, 8, and 9, how many 7-digit numbers can be constructed if the number must begin with an odd digit and digits may not be repeated?
The required numbers that can be constructed are given as 1440 words.
Given that,
Using the digits 3, 4, 5, 6, 7, 8, and 9, how many 7-digit numbers can be constructed if the number must begin with an odd digit and digits may not be repeated is to be determined.
In arithmetic, combination and permutation are two different ways of grouping elements of a set into subsets. In combination, the components of the subset can be recorded in any order. In a permutation, the components of the subset are listed in a distinctive order.
Here,
Because, there are four odd digits, there are only four options for the initial digit.
It has six digits from which to choose 4, 6C4 = 15. And there are 4! = 24 different ways to arrange those four digits.
The total number of choices is 4 × 15 × 24 = 1440
Thus, the required numbers that can be constructed are given as 1440 words.
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A small business owner made $40,000 the first year he owned his store and made an additional 7% over
the previous year in each subsequent year. Find how much he made during his fourth year of business.
Find his total earnings during the first four years.
The small business owner made $44,486 during his fourth year of business. His total earnings during the first four years equal $173,168.
The small business owner made $44,486 during his fourth year of business.
To calculate this, we find that he made $40,000 in the first year, then 7% more each year.
So, in the fourth year:
[tex]$40,000 + ( 0.07 * 40,000) + (0.07^2 * 40,000) + (0.07^3 * 40,000) = 44,486[/tex]
His total earnings during the first four years equal $173,168.
This total is found by summing his earnings from each year:
[tex]$40,000 + 42,800 + 45,796 + 44,486 = 173,168.[/tex]
The number of automobiles sold weekly at a certain dealership is a random variable with expected value 16 .Give an upper bound to the probability that
(a) next week's sales exceed 18;
(b )next week's sales exceed 25.
(c) Suppose that the variance of the number of automobiles sold weekly is 9. Give a lower bound to the probability that next week's sales are between 10 and 22 inclusively;
(d) give an upper bound to the probability that next week's sales exceed 18.
Using the properties of normally distributed random variables and Chebyshev's inequality, we can find the upper and lower bounds for the probability of next week's automobile sales, given the expected value and variance. These calculations involve finding the 'k' value, which is derived from subtracting the expected value from the target sales value, and then using this in the inequality.
Explanation:The subject of this question is Probability in Mathematics, and this question is likely for a College level course. To bound the probability of a normally distributed random variable exceeding a given value, we can use Chebyshev's inequality: Pr(|X - μ| ≥ kσ) ≤ 1 / k2. Here, X represents the random variable (automobile sales), μ is its expected value (16 sales), and σ is its standard deviation (the square root of the variance).
To find an upper bound for the probability that next week's sales exceed 18, we calculate k as |18 - 16| and use the inequality to calculate the upper bound. The same method applies for finding an upper bound that next week's sales exceed 25. For the probability that next week's sales are between 10 and 22, we have two k-values (|10-16| and |22-16|), and we use both in the inequality to find the lower bound. The fourth part of the question appears to be a repeat of part (a).Learn more about Probability here:
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A bank is offering 2.5% simple interest on a savings account. If you deposit $5000, how much interest will you earn in one year?
Answer:
125.00
Step-by-step explanation:
The senate in a certain state is comprised of 51 republicans, 48 democrats, and 1 independent. How many committees can be formed if each committee must have 3 republicans and 2 democrats
There are 23,490,600 committees that can be formed if each committee must have 3 Republicans and 2 Democrats
Used the concept of combination which states that,
A combination is a technique to determine the number of possible arrangements in a collection of items where the order of the selection does not matter.
Given that,
The Senate in a certain state is comprised of 51 Republicans, 48 Democrats, and 1 independent.
Since we have 51 Republicans,
So we can choose 3 Republicans out of 51 in [tex]^{51} C_3[/tex] ways.
Similarly, we have 48 Democrats, so we can choose 2 Democrats out of 48 in [tex]^{48} C_2[/tex] ways.
Hence, the number of committees:
[tex]^{51} C_3 \times ^{48} C_2[/tex]
Using the formula for combinations,
[tex]^{n} C_r = \dfrac{n!}{r! (n - r)!}[/tex]
[tex]= \dfrac{51!}{3!(51 - 3)!} \times \dfrac{48!}{2!(48 - 2)!}[/tex]
[tex]= \dfrac{51!}{3!(48)!} \times \dfrac{48!}{2!(46)!}[/tex]
[tex]= \dfrac{51 \times 50 \times 49 \times 48!}{3!(48)!} \times \dfrac{48\times 47 \times 46!}{2!(46)!}[/tex]
[tex]= \dfrac{51 \times 50 \times 49 }{3!} \times \dfrac{48\times 47 }{2!}[/tex]
[tex]= 20,825 \times 1,128[/tex]
So, the number of committees is,
[tex]= 23,490,600[/tex]
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The sum of five times a number and 6 more than the number is the same as the difference between -18 and twice the number. What is the number?
The number is given by -3.
What is an Equation?Equation is a mathematical relation between more than one variable, number or mathematical quantities involving mathematical operations using equal sign in between them.
Let the number be x
Five times the number is = 5x
6 more than the number is = x+6
Sum of five times a number and 6 more than the number is = 5x+(x+6) = 5x+x+6 = 6x+6
The difference between -18 and twice the number = -18-2x
According to condition,
6x+6 = -18-2x
6x+2x = -18-6
8x = -24
x = -24/8 = -3
Hence the number is -3.
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A popular extreme sport is artificial wall climbing. A climber is resting at a height of 42 feet while on a climbing tower. If this is 60% of the tower's total height, find the height of the tower.
The total height of the climbing tower is found by dividing 42 feet by 0.60, as 42 feet represents 60% of the total height. The calculation gives us 70 feet, which is the height of the tower.
The student has asked a question related to artificial wall climbing and its connection to mathematics, specifically regarding percentages and solving for an unknown total given a part and its percentage of the whole. The question asks to find out the total height of a climbing tower given that 42 feet is 60% of its total height.
To solve this, we would set up a proportion where 42 feet equals 60% of the total height. Using the formula part = percent × whole, we can represent the total height of the tower by the variable x. Therefore, the equation is:
42 = 0.60x
To find x, we divide both sides of the equation by 0.60:
x = 42 / 0.60
x = 70 feet
Thus, the height of the tower is 70 feet.
If a circular property has a diameter of 50' and costs $120 per square foot, what is the cost of the property?
To calculate the cost of a circular property with a diameter of 50', determine the area using the formula π * r^2, and then multiply by the cost per square foot. The total cost of the property comes to $235,619.40.
Explanation:The question asks us to calculate the cost of a property with a circular shape having a diameter of 50'. To begin with, we will find the area of the circular property by using the formula for the area of a circle, A = π * r2, where π (pi) is approximately 3.14159, and r is the radius of the circle. Since the diameter is given as 50', the radius (r) would be half of that, which is 25'. Therefore, the area (A) of the property is 3.14159 * (25')2 = 3.14159 * 625 = 1963.495 square feet.
Now, if the cost is $120 per square foot, we multiply the area by this cost to find the total price of the property. So, the total cost is 1963.495 square feet * $120/square foot = $235,619.40.
Therefore, the cost of the property is $235,619.40.
rahms spent two and a half hours writing an essay it took him 4 times as long to finish his science project how long did it take rahm to write the essay and finish the science project
the Golden Ratio, is present not just in mathematics, but may also be present within your own brain and body (at the atomic or subatomic level), what do you think it means? Is this number evidence of a grand design, a massive freak coincidence or something else?
Sisters Corp expects to earn $5 per share next year. The firm's ROE is 14% and its plowback ratio is 60%. If the firm's market capitalization rate is 10%. a. Calculate the price with the constant dividend growth model. (Do not round intermediate calculations.) Price $ ____? b. Calculate the price with no growth. Price $ _____? c. What is the present value of its growth opportunities? (Do not round intermediate calculations.) PVGO $ ______?
To calculate the price using the constant dividend growth model, we can use the formula: Price = Dividend / (Market Capitalization Rate - Dividend Growth Rate). Given the expected dividend per share, market capitalization rate, ROE, and plowback ratio, we can calculate the price per share.
Explanation:To calculate the price using the constant dividend growth model, we can use the formula:
Price = Dividend / (Market Capitalization Rate - Dividend Growth Rate)
Given that the expected dividend per share is $5 and the market capitalization rate is 10%, we can calculate the dividend growth rate using the ROE and plowback ratio. The plowback ratio is the portion of earnings retained by the company to finance growth. It is calculated as (1 - Dividend Payout Ratio). The dividend payout ratio is the portion of earnings paid out as dividends, which is the complement of the plowback ratio.
Using the formula for the dividend growth rate:
Dividend Growth Rate = ROE * Plowback Ratio
Given that the ROE is 14% and the plowback ratio is 60%, the dividend growth rate is:
Dividend Growth Rate = 14% * 60% = 8.4%
Now we can calculate the price:
Price = $5 / (10% - 8.4%) = $5 / 1.6% = $312.50 per share
Law of cosines: a2 = b2 + c2 – 2bccos(A)
Which equation correctly uses the law of cosines to solve for y?
92 = y2 + 192 – 2(y)(19)cos(41°)
y2 = 92 + 192 – 2(y)(19)cos(41°)
92 = y2 + 192 – 2(9)(19)cos(41°)
y2 = 92 + 192 – 2(9)(19)cos(41°)
A real estate agent makes an annual salary of $30,000 plus a 4% commission on sales. If the realtor's salary is $84,000, how much did he make in sales?
He made 1,350,000 in sales.
What is percentage?A percentage is a number that tells us how much out of 100.
Given that, A real estate agent makes an annual salary of $30,000 plus a 4% commission on sales, and a realtor's salary is $84,000,
84000 - 30000=54000
54000 is his sales
100/4 = 25
54000*25 = 1350000
1,350,000
Hence, He made 1,350,000 in sales.
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if 3 apples cost $1.25 what is the cost of 7 apples?
a) A box contains 50 diodes of which 10 are known to be bad. A diode is selected at random. What...
a) A box contains 50 diodes of which 10 are known to be bad. A diode is selected at random. What is the probabil ity that it is bad?
b) If the first diode drawn from the box was good, what is the probability that a: second diode drawn wi ll be good?
c) If two diodes are drawn from the box what is the probability that they are both good?
Final answer:
a) The probability of selecting a bad diode is 0.2 (20%). b) If the first diode is good, the probability of the second diode being good is 0.816 (81.6%). c) The probability that both diodes are good is 0.6326 (63.26%).
Explanation:
a) Probability that the diode is bad:
There are 50 diodes in total, with 10 known to be bad. The probability of selecting a bad diode is calculated by dividing the number of bad diodes by the total number of diodes: 10/50 = 1/5 = 0.2 or 20%.
b) Probability of second diode being good:
If the first diode drawn is good, there will be 49 diodes left, with 9 being bad. The probability of selecting a good diode as the second one is 40/49 = 0.816 or 81.6%.
c) Probability that both diodes are good:
If two diodes are drawn, there will be 48 diodes left, with 9 being bad. The probability of selecting a good diode for the first draw is 40/50 and for the second draw is 39/49. Therefore, the probability of both diodes being good is (40/50) * (39/49) = 0.6326 or 63.26%.
The probability of selecting a bad diode from the box of 50 is 0.2. If a good diode is selected first, the probability of selecting another good diode is 39/49. The probability of selecting two good diodes consecutively is approximately 0.6367.
Explanation:The question deals with the concept of probability within a discrete random variable scenario. Specifically, it requires calculating the probabilities of selecting good and bad diodes from a batch with a known distribution of working and defective parts.
a) The probability that a randomly selected diode is bad is the number of bad diodes divided by the total number of diodes. Therefore, it is 10/50 or 1/5 or 0.2.b) If the first diode drawn is good, we have 49 diodes left with 10 bad still among them. The probability that a second diode drawn will be good is the number of remaining good diodes divided by the total remaining diodes, which is 39/49.c) To find the probability that both diodes are good when two are drawn without replacement, we multiply the probability of drawing one good diode by the probability of drawing a second good diode (having already drawn one good). This gives us (40/50) * (39/49), which simplifies to 0.8 * 0.7959 or approximately 0.6367.6/15=2/c need to solve
The right triangle ABC shown below is inscribed inside a parabola. Point B is also the maximum point of the parabola (vertex) and point C is the x intercept of the parabola. If the equation of the parabola is given by y = -x2 + 4x + C, find C so that the area of the triangle ABC is equal to 32 square units.
Testing a Lie Detector The table below includes results from polygraph (lie detector) experiments conducted by researchers Charles R. Honts (Boise State University) and Gordon H. Barland (Department of Defense Polygraph Institute). In each case, it was known if the subject lied or did not lie, so the table indicates when the polygraph test was correct. Use a 0.05 significance level to test the claim that whether a subject lies is independent of the polygraph test indication. Do the results suggest that polygraphs are effective in distinguishing between truths and lies?
Did the Subject Actually Lie?
Did the Subject Actually Lie?
No (Did Not Lie)
Yes (Lied)
Polygraph test indicated that the subject lied.
15
42
Polygraph test indicated that the subject did not lie.
To test the claim that whether a subject lies is independent of the polygraph test indication, a chi-square test of independence is conducted. The results, with a chi-square value of 34.774, suggest that polygraphs are effective in distinguishing between truths and lies.
Explanation:To test the claim that whether a subject lies is independent of the polygraph test indication, we will conduct a chi-square test of independence. Using a significance level of 0.05, we will compare the observed frequencies in the table to the expected frequencies assuming independence. If the test statistic is significant, it would suggest that polygraphs are effective in distinguishing between truths and lies.
First, let's calculate the expected frequencies assuming independence. The total number of subjects who lied is 42+15=57, and the total number of subjects who did not lie is 15+129=144. The proportion who lied is 57/(57+144)=0.283. Multiplying this proportion by the number of subjects who were indicated as lying (57), we would expect (0.283)(57)=16.131 subjects to be accurate. Similarly, multiplying the proportion who did not lie by the number of subjects who were indicated as not lying, we would expect (0.717)(144)=103.488 subjects to be accurate.
Now, let's calculate the chi-square test statistic. Using the formula: chi-square = Σ (observed frequency - expected frequency)^2 / expected frequency, we calculate the chi-square value to be 34.774. Comparing this value to the critical chi-square value at a significance level of 0.05, with (2-1)(2-1) = 1 degree of freedom, the critical chi-square value is 3.841. Since the calculated chi-square value is greater than the critical value, we reject the null hypothesis and conclude that the results suggest that polygraphs are effective in distinguishing between truths and lies.
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What is 1/6+1/12=what does this equal?
a small pump can drain a pool in 9 hours. a large pump could drain the same pool in 7 hours. how long will it take to drain the pool of both pumps are used simultaneously
The combined rate of the small and large pumps is 16/63 pools per hour, and it will take them approximately 3.94 hours to drain the pool when working together.
Explanation:To figure out how long it would take for both pumps to drain the pool simultaneously, we will use the concept of rates. Since the small pump can drain the pool in 9 hours, it has a rate of 1/9 pools per hour. Similarly, the large pump can drain the pool in 7 hours, so it has a rate of 1/7 pools per hour.
When we use both pumps together, we add their rates. So, the combined rate of the small and large pumps is (1/9 + 1/7) = 16/63 pools per hour.
To find the time it will take them to drain the pool together, we divide the size of the pool (1 pool) by their combined rate (16/63 pools per hour). Therefore, it will take the two pumps approximately 3.94 hours to drain the pool when working together.
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14.4% of 72.5 is what number?
Final answer:
To find 14.4% of 72.5, convert the percentage to a decimal and multiply it by the number, resulting in 10.44.
Explanation:
To calculate 14.4% of 72.5, you can use the following steps:
Convert the percentage to a decimal by dividing by 100: 14.4% ÷ 100 = 0.144.Multiply the decimal by the number: 0.144 × 72.5.Perform the multiplication: 0.144 × 72.5 = 10.44.Therefore, 14.4% of 72.5 is 10.44.
Beatrice invests $1,440 in an account that pays 3 percent simple interest. How much more could she have earned over a 4-year period if the interest had compounded annually?
$7.93
$31.73
$17.61
$28.37
$21.28
To calculate the compound interest, use the formula A = P(1 + r/n)^(n*t), where A is the future amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. Beatrice could have earned $419.38 more over a 4-year period with compounded interest.
Explanation:To calculate the compound interest, we can use the formula: A = P(1 + r/n)^(n*t), where A is the future amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. In this case, Beatrice invests $1,440 at an interest rate of 3% for 4 years.
Using the formula, the future amount with compound interest would be: A = 1440 * (1 + 0.03/1)^(1*4) = $1576.18.
To find the difference in earnings, we subtract the future amount with simple interest from the future amount with compound interest: $1576.18 - $1156.80 = $419.38. Therefore, Beatrice could have earned $419.38 more over a 4-year period with compounded interest.
A bag contains
9
marbles:
2
are green,
2
are red, and
5
are blue. Omar chooses a marble at random, and without putting it back, chooses another one at random. What is the probability that the first marble is green and the second is red? Write your answer as a fraction in simplest form.