Answer:
less than 3
Step-by-step explanation:
In terms of x, find an expression that represents the area of the shaded region. The outer square has side lengths of (x+5) and the inner square has side lengths of (x-2), as shown.
Answer:
= (x+5)² = x² + 10x + 25
= (x-2)² = x² - 4x + 4
= (x² + 10x + 25) - (x² - 4x + 4)
= x² + 10x + 25 - x² + 4x - 4
= 14x + 21 square units
The area of the shaded region is found by subtracting the area of the inner square, (x-2)², from the area of the outer square, (x+5)², resulting in the expression 14x + 21.
Explanation:The area of the shaded region in this problem represents the difference between the area of the outer square and the inner square.
To find this, we calculate the area of each square individually and then subtract one from the other.
First, the area of the outer square is (x+5)² and the area of the inner square is (x-2)².
Now, we find the difference between these two areas to isolate the shaded region:
Area of shaded region = (x+5)² - (x-2)²
To expand this, we use the binomial expansion:
(x+5)² = x² + 10x + 25(x-2)² = x² - 4x + 4Now we subtract the smaller area from the larger area:
Shaded region = (x² + 10x + 25) - (x² - 4x + 4)
Shaded region = x² + 10x + 25 - x² + 4x - 4
Shaded region = 14x + 21
This expression represents the area of the shaded region in terms of x.
Consider two sizes of disk, both of mass M. One size of disk has radius R; the other has radius 4R. System A consists of two of the larger disks rigidly connected to each other with a common axis of rotation. System B consists of one of the larger disks and a number of the smaller disks rigidly connected with a common axis of rotation. If the moment of inertia for system A = the moment of inertia for system B, how many of the smaller disks are in system B? 1 4 10 16
Answer:
16
Step-by-step explanation:
Moment of inertia of a disk is proportional to its mass and to the square of its radius. For two disks with the same mass, the larger one will have a moment of inertia that is (4R/R)^2 = 16 times that of the smaller one.
It will take 16 smaller disks to make the systems have the same moment of inertia.
What is the GCF of the expression a2b2c2 + a2bc2 - a2b2c
Answer:
a^2bc
Step-by-step explanation:
The GCF of the expression a2b2c2 + a2bc2 - a2b2c is a2bc.
The greatest common factor (GCF) of an algebraic expression is the largest polynomial that divides each of the terms without leaving a remainder. To find the GCF of the expression a2b2c2 + a2bc2 - a2b2c, first identify the common factors in each term.
Inspecting each term we see that a2 is a common factor for all of them, and the smallest power of b and c present in all terms is b and c, respectively. Therefore, the GCF is a2bc.
What are the solutions of the equation x4 + 6x2 + 5 = 0? Use u substitution to solve.
Answer:
2nd answer.
Step-by-step explanation:
see attached.
Answer with Step-by-step explanation:
We have to find the solution of the equation:
[tex]x^4+6x^2+5=0[/tex]
Let u=x²
Then, above equation is transformed to:
[tex]u^2+6u+5=0[/tex]
it could also be written as:
[tex]u^2+5u+u+5=0[/tex]
u(u+5)+1(u+5)=0
(u+1)(u+5)=0
either u+1=0 or u+5=0
either u= -1 or u= -5
Putting u=x²
x² = -1 or x² = -5
On taking square root both sides
x= ± i or x= ± i√5
Hence, roots of the equation [tex]x^4+6x^2+5=0[/tex] are:
i , -i , i√5 and -i√5
Compute the face value of a 90-day promissory note dated October 22, 2018 that has a maturity value of $76,386.99 and an interest rate of 7.5% p.a.
Answer:
The face value would be $75,000
Step-by-step explanation:
Maturity value = $76,386.99
Time = 90 days
Rate of interest = 7.5%
Let face value be 'x'
By using the formula [tex]A=P(1+\frac{RT}{100})[/tex]
$76,386.99 = [tex]x(1+\frac{7.5\times \frac{90}{365}}{100})[/tex]
Time in years = [tex]\frac{90}{365}[/tex]
⇒ $76,386.99 = x( 1 + 0.01849315 )
⇒ x = [tex]\frac{76,386.99}{1.01849315}[/tex]
x = $75,000
The face value would be $75,000
How many mL of 75% alcohol should be mixed with 10% of 1000 cc alcohol to prepare 30% of 500 mL alcohol solution? a. 346.16 mL b. 234.43 mL c. 153.84 mL d. 121.12 mL e.
Answer:
C. 153.84 mL
Step-by-step explanation:
Let's say x is the volume of 75% solution and y is the volume of 10% solution.
Sum of the volumes:
x + y = 500
Sum of the alcohol amounts:
0.75x + 0.10y = 0.30(500)
0.75x + 0.10y = 150
Solve the system of equations using either substitution or elimination. I'll use substitution.
y = 500 - x
0.75x + 0.10 (500 - x) = 150
0.75x + 50 - 0.10x = 150
0.65x = 100
x = 153.84
You need 153.84 mL of 75% solution.
"153.84 mL" of 75% alcohol should be added. A further explanation is provided below.
Let,
75% alcohol used be "x".10% alcohol used be "y".then,
→ [tex]x+y = 500[/tex]
[tex]y = (500-x)[/tex]
now,
→ [tex]75(x)+10(500-x) = 500\times 30[/tex]
[tex]65x+5000=15000[/tex]
[tex]65x=15000-5000[/tex]
[tex]65x=10000[/tex]
[tex]x = \frac{10000}{65}[/tex]
[tex]= \frac{2000}{13}[/tex]
[tex]= 153.84 \ mL[/tex]
Thus the above response i.e., "option c" is correct.
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which of the following is the quotient of .4375 divided by .35
Answer:
Quotient will be 1.25
Step-by-step explanation:
First we convert decimal numbers to fractions. So write down the decimal divided by 1 and then multiply both top and bottom with 10 for every number after decimal point.
Here we found for .4375 = [tex]\frac{4375}{10000}[/tex]
and .35 = [tex]\frac{35}{100}[/tex]
Now we divide both the numbers as
= [tex]\frac{\frac{4375}{1000} }{\frac{35}{100} }[/tex]
= [tex]\frac{4375}{1000}[/tex] × [tex]\frac{100}{35}[/tex]
= [tex]\frac{125}{100}[/tex]
= 1.25
Quotient will be 1.25
Final answer:
The quotient of 0.4375 divided by 0.35 is 1.25, which rounded to the tenths place is 1.3.
Explanation:
The student is asking to find the quotient of two decimal numbers, which is a basic arithmetic operation involving division. The numbers are 0.4375 and 0.35. To find the quotient, simply divide 0.4375 by 0.35.
Using a calculator or performing the division manually, you would proceed as follows:
Adjust the decimals by multiplying both numbers by 100 to make them whole numbers, resulting in 43.75 divided by 35.
Perform the division to get the preliminary result: 43.75 / 35 = 1.25.
Since we need to round the final answer to the tenths place based on the least precise number given (35.5 g), round 1.25 to one decimal place, which is 1.3 (1.25 rounds up because the next digit, 5, is equal to or greater than 5).
Therefore, the quotient of 0.4375 divided by 0.35, rounded to the tenths place, is 1.3.
If you drive 5 miles south, then make a left turn and drive 12 miles east, how far are you, in a straight line, from your starting point? Use the Pythagorean Theorem to solve the problem. Use a calculator to find square roots, rounding to the nearest tenth as needed.
Answer: Hence, the distance covered in a straight line from the starting point is 13 miles.
Step-by-step explanation:
Since we have given that
Distance between AB = 5 miles
Distance between BC = 12 miles
We need to find the distance covered from the starting point.
We will use "Pythagorean Theorem":
[tex]H^2=P^2+B^2\\\\AC^2=AB^2+BC^2\\\\AC^2=5^2+12^2\\\\AC^2=25+144\\\\AC^2=169\\\\AC=\sqrt{169}\\\\AC=13\ miles[/tex]
Hence, the distance covered in a straight line from the starting point is 13 miles.
find the solutions of the system
y=x^2+3x-4
y=2x+2
a. (-3,6) and (2,-4)
b. (-3,-4) and (2,6)
c. (-3,-4) and (-2,-2)
d. no solution
Answer:
b. (-3, -4) and (2, 6)
Step-by-step explanation:
By the transitive property of equality, if y equals thing 1 and y also equals thing 2, then thing1 and thing 2 are also equal. So we will set them equal to each other and factor to solve for the 2 values of x:
[tex]2x+2=x^2+3x-4[/tex]
Get everything on one side of the equals sign, set the whole mess equal to 0, and combine like terms to get:
[tex]0=x^2+x-6[/tex]
Because this is a second degree polynomial, a quadratic to be precise, it has 2 solutions. We need to find those 2 values of x and then use them in either one of the original equations to solve for the y values that go with each x.
Factoring that polynomial above gives you the x values of x = -3 and 2. Sub in -3 first:
y = 2(-3) + 2 and
y = -6 + 2 so
y = -4
Therefore, the coordinate is (-3, -4).
Onto the next x value of 2:
y = 2(2) + 2 and
y = 4 + 2 so
y = 6
Therefore, the coordinate is (2, 6)
15, Evaluate 6 choose 4.
Answer: The required result is 15.
Step-by-step explanation: We are given to evaluate the following :
"6 choose 4".
Since we are to choose 4 from 6, so we have to use the combination of 6 different things chosen 4 at a time.
We know that
the formula for the combination of n different things chosen r at a time is given by
[tex]^nC_r=\dfrac{n!}{r!(n-r)!}.[/tex]
For the given situation, n = 6 and r = 4.
Therefore, we get
[tex]^6C_4=\dfrac{6!}{4!(6-4)!}=\dfrac{6!}{4!2!}=\dfrac{6\times5\times4!}{4!\times2\times1}=15.[/tex]
Thus, the required result is 15.
Out of 25 attempts, a basketball player scored 17 times. One-half of the missed shots are what % of the total shots?
Answer:
16%
Step-by-step explanation:
Eight shots were missed. Take half of eight; 4. You now have 4\25, which is 160‰ [16%].
Answer:
%16
Step-by-step explanation:
Step 1: Find the shots missed
25 - 17 = 8
Step 2: Find half of the shots missed
8 / 2 = 4
Step 3: Divide 4 by 25
4/25 = 0.16
Step 4: Convert to Percent
0.16 * 100 = %16
Answer: %16
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A packet of sour worms contains four strawberry, four lime, two black currant, two orange sour, and three green apple worms. What is the probability that Dustin will choose a green apple sour worm, P(green apple)?
Answer:
3/15 or 0.2
Step-by-step explanation:
Do more Republicans (group A) than Democrats (group B) favor a bill to make it easier for someone to own a firearm? Two hundred Republicans and two hundred Democrats were asked if they favored a bill that made it easier for someone to own a firearm. How would we write the alternative hypothesis?
The alternative hypothesis would state that the proportion of Republicans who favor a bill to make gun ownership easier is not equal to the proportion of Democrats who favor the same.
Explanation:The question was regarding how to construct an alternative hypothesis for a study on political beliefs and opinions on firearm ownership. In this case, the alternative hypothesis statement goes against the null hypothesis. The null hypothesis would be that there's no significant difference between the proportions of Republicans and Democrats that favor a bill making gun ownership easier. So, the alternative hypothesis can be written as: 'The proportion of Republicans (Group A) who favor a bill making it easier for someone to own a firearm is not equal to the proportion of Democrats (Group B) who favor the same.'
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The alternative hypothesis can be written as: H_A: The proportion of Republicans who favor a bill to make it easier for someone to own a firearm differs from the proportion of Democrats who favor the same.
Explanation:The alternative hypothesis can be written as:
HA: The proportion of Republicans who favor a bill to make it easier for someone to own a firearm differs from the proportion of Democrats who favor the same.
Alternatively, it can be written as:
HA: pA ≠ pB
where pA is the proportion of Republicans who favor the bill and pB is the proportion of Democrats who favor the bill.
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2x - 20 = 32
20 - 3x = 8
6x - 8 = 16
-13 - 3x = -10
Answer:
Step-by-step explanation:
1st one is x=26
2nd one is x=4
3rd is x=4
4th is x=-1
Hope that helps!
Answer:
so the answers are 26, 4, 4, and -1
Step-by-step explanation:
If you want me to solve all of them it is: Your getting x by itself
so do the opposite of each problem i'll do the first one
2x - 20 = 32
+ 20 +20
2x = 52 divide the 2
2 2
x = 26
Hope my answer has helped you if not i'm sorry.
There are red blood cells contained in 50 oubic millimeters of blood se scientific notation. Use the multiplication symbol in the math palette as needed )
Answer: 5\times10
Step-by-step explanation:
We know that the scientific notation is a representation of a very large or a very small number in the product of a decimal form of number (commonly between 1 and 10) and powers of ten.
Given : There are red blood cells contained in 50 cubic millimeters of blood .
The representation of 50 cubic millimeters in scientific notation is given by :-
[tex]5\times10\ \text{cubic millimeters }[/tex]
In January 2013 a country‘s first class mail rates increased to 42 cents for the 1st ounce and 22 cents for each additional ounce. Is Sabrina spent $16.24 for a total of 52 stamps of these two denominations how many stamps of each denomination did she buy?
She bought ___ 42 cent stamps
And _____ 22 cent stamps
Answer:
She bought 24 42-cent stampsAnd 28 22-cent stampsStep-by-step explanation:
Let n represent the number of 42-cent stamps Sabrina bought. Then 52-n is the number of 22-cent stamps she bought. Her total expense was ...
0.42n +0.22(52 -n) = 16.24 . . . . total price of stamps
0.20n + 11.44 = 16.24 . . . . . . . . . simplify
0.20n = 4.80 . . . . . . . . . . . . . . . . subtract 11.44
n = 24 . . . . . . . . . . . . . . . . . . . . . . divide by the coefficient of n
52-n = 28 . . . . . . . . . . . . . . . . . . . find the number of 22-cent stamps
She bought 24 42-cent stamps and 28 22-cent stamps.
She bought 24-42 cent stamps
And, 28-22 cent stamps.
Calculation of number of stamps:Here we assume n be the number of 42-cent stamps
The equation should be
0.42n +0.22(52 -n) = 16.24
0.20n + 11.44 = 16.24
0.20n = 4.80
n = 24
Now
= 52 - n
= 52 - 24
= 28
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Evaluate the Expression B^2-4 ac given by that a = -2 ,, b= -2 and c =2
F* you B*!!!!!! Your so S*! That's the easiest thing in the world!!
This year, Druehl, Inc., will produce 57,600 hot water heaters at its plant in Delaware, in order to meet expected global demand. To accomplish this, each laborer at the plant will work 160 hours per month. If the labor productivity at the plant is 0.15 hot water heaters per labor hour, how many laborers are employed at the plant?
Answer:
200
Step-by-step explanation:
Goal 57600 heaters per year
160 hr per 1 month
so 160(12)hr per 1 year
that is 1920 hr per 1 year
We also have that .15 heaters are produced every 1 hour
so multiply 1920 by .15 and you have your answer
160(12)(.15)=288 heaters are produced per one person per year
so we need to figure how many people we need by dividing year goal by what one person can do
57600/288=200 people needed
200 laborers are employed at the plant.
First find out the number of hours each worker will have to work in a year:
= Number of hours per month x 12 months
= 160 * 12
= 1,920 hours
Find out the number of units each worker will produce in those hours:
= Annual number of hours x Units per hour
= 1,920 * 0.15
= 288 heaters
The number of laborers employed is:
= Yearly demand of heaters / Number of heaters produced per worker
= 57,600 / 288
= 200 laborers
The plant employs 200 laborers.
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Find the geometric means in the following sequence.
Answer:
Choice A
Step-by-step explanation:
a=-6 (1st term)
ar= (2nd term)
ar^2= (3rd term)
ar^3 (4th term)
ar^4= (5th term)
ar^5=-1458 (6th term)
a=-6 so -6r^5=-1458
divide both sides by -6 giving r^5=243 so to obtain r you do the fifth root of 243 which is 3.
The common ratio is 3.
so ar=6(-3)=-18 (2nd term)
Only choice A fits this.
In 1987, the General Social Survey asked, "Have you ever been active in a veteran's group? " For this question, 52 people said that they did out of 98 randomly selected people. The General Social survey randomly selects adults living in the US. Someone wanted to compute a 95% confidence interval for p. What is parameter?
Final answer:
The parameter in this question refers to the population proportion. To compute a 95% confidence interval for the proportion, you can use the formula: p ± z × √(p × (1-p) / n). The sample proportion is 0.53 and the sample size is 98. By plugging these values into the formula, you can calculate the confidence interval.
Explanation:
The parameter in this question refers to the population proportion. In statistics, a parameter is a measure that describes a characteristic of a population. In this case, the parameter is the proportion of all adults living in the US who have been active in a veteran's group. To compute a 95% confidence interval for this proportion, you can use the formula: p ± z × √(p × (1-p) / n), where p is the sample proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size.
Using the provided information, the sample proportion is 52/98 = 0.53. To find the z-score for a 95% confidence level, you can use a standard normal distribution table or a calculator with the function invNorm(0.975). The z-score for a 95% confidence level is approximately 1.96. The sample size is 98. Plugging these values into the formula, you can calculate the confidence interval for the population proportion.
Confidence interval = 0.53 ± 1.96 × √(0.53 × (1-0.53) / 98) = 0.53 ± 0.0907
The parameter p is the true proportion of adults in the US who have ever been active in a veteran's group, and the 95% confidence interval for this parameter is (0.4317, 0.6295).
The formula for a 95% confidence interval for a proportion is given by:
[tex]\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \][/tex]
where z is the z-score corresponding to the desired confidence level. For a 95% confidence interval, the z-score is approximately 1.96.
Let's calculate the confidence interval:
1. Calculate the sample proportion [tex]\( \hat{p} \)[/tex]:
[tex]\[ \hat{p} = \frac{52}{98} \approx 0.5306 \][/tex]
2. Calculate the standard error of the proportion:
[tex]\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.5306(1 - 0.5306)}{98}} \approx \sqrt{\frac{0.2503}{98}} \approx \sqrt{0.002554} \approx 0.0505 \][/tex]
3. Find the z-score for a 95% confidence interval, which is approximately 1.96.
4. Calculate the margin of error:
[tex]\[ ME = z \times SE \approx 1.96 \times 0.0505 \approx 0.0989 \][/tex]
5. Calculate the confidence interval:
[tex]\[ \text{Lower bound} = \hat{p} - ME \approx 0.5306 - 0.0989 \approx 0.4317 \] \[ \text{Upper bound} = \hat{p} + ME \approx 0.5306 + 0.0989 \approx 0.6295 \][/tex]
Therefore, the 95% confidence interval for the proportion p of all adults living in the US who have ever been active in a veteran's group is approximately (0.4317, 0.6295).
Translate the Variable Expression 3n -7 into Verbal Expression
Step-by-step explanation:
[tex]3n-7\\\\\text{The difference between three times the number n and seven.}[/tex]
An expression is a set of numbers, variables, and mathematical operations. The Variable Expression 3n -7 into Verbal Expression can be written as expression 7 less than 3 times a number 'n'.
What is an Expression?In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.
The expression that is given to us is 3n -7, this expression can be written as a verbal expression 7 less than 3 times a number 'n' or 7 subtracted from 3 times of number 'n'.
Hence, the Variable Expression 3n -7 into Verbal Expression can be written as expression 7 less than 3 times a number 'n'.
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Let F = (z − y) i + (x − z) j + (y − x) k . Let C be the rectangle of width 2 and length 5 centered at (7, 7, 7) on the plane x + y + z = 21, oriented clockwise when viewed from the origin. (a) Find curlF . curlF = ⟨2,2,2⟩ (b) Use Stokes' Theorem to find F · dr C . F · dr C = −60 √3
The curl of the vector field F is 2i + 2j + 2k. The dot product of F and dr along the closed path C is -60√3.
Explanation:To find the curl of vector field F, we need to compute the partial derivatives of its components with respect to x, y, and z. In this case, F = (z-y)i + (x-z)j + (y-x)k. Taking the partial derivatives, we get curlF = 2i + 2j + 2k.
The dot product of F and dr along the closed path C can be calculated using Stokes' Theorem. By evaluating the dot product and integrating over C, we find that F · dr = -60√3.
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The unemployment rate in a city is 13%. If 6 people from the city are sampled at random, find the probability that at least 3 of them are unemployed. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. (If necessary, consult a list of formulas.)
The weights of broilers (commercially raised chickens) are approximately normally distributed with mean 1387 grams and standard deviation 161 grams. What is the probability that a randomly selected broiler weighs more than 1,425 grams?
Answer: 0.3936
Step-by-step explanation:
Given: Mean : [tex]\mu =1387 \text{ grams}[/tex]
Standard deviation : [tex]\sigma = 161 \text{ grams}[/tex]
The formula to calculate z is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 1,425 grams
[tex]z=\dfrac{1425-1387}{161}=0.23602484472\approx0.27[/tex]
The P Value =[tex]P(X>1425)=P(z>0.27)=1-0.6064198=0.3935802\approx0.3936[/tex]
Hence, the probability that a randomly selected broiler weighs more than 1,425 grams =0.3936
Final Answer:
There is approximately a 40.66% chance that a randomly selected broiler weighs more than 1,425 grams.
Explanation:
To solve this problem, you will need to apply the properties of the normal distribution. We want to find out the probability that a broiler weighs more than 1,425 grams.
Given:
- Mean (μ) = 1387 grams
- Standard deviation (σ) = 161 grams
- X = 1425 grams (the value we're interested in)
Step 1: First, we compute the z-score for the weight of 1425 grams. The z-score is a measure of how many standard deviations an element is from the mean. It can be calculated using the formula:
[tex]\[ z = \frac{(X - \mu)}{\sigma} \][/tex]
where X is the value for which we're finding the probability, μ is the mean, and σ is the standard deviation.
Step 2: Insert the values into the formula to compute the z-score for 1425 grams:
[tex]\[ z = \frac{(1425 - 1387)}{161} \\\\\[ z = \frac{38}{161} \\\\\[ z \approx 0.236 \][/tex]
Step 3: Once we have the z-score, we can use the z-table (a standard normal distribution table) to find out the probability of a z-score being less than 0.236. However, since we want the probability that the broiler weighs more than 1425 grams, we are interested in the probability of a z-score being greater than 0.236.
Step 4: Look up the corresponding probability for z = 0.236 on the z-table. The z-table gives us the area under the normal curve to the left of the z-score.
Let's assume the z-table gives us a probability of P(Z < 0.236). The value would typically be around 0.5934, which means there is a 59.34% chance that a random broiler will weigh less than 1425 grams.
Step 5: To find the probability that a broiler weighs more than 1425 grams, we subtract the value found in the z-table from 1 because the total area under the curve equals 1 (which corresponds to the probability of all possible outcomes).
[tex]\[ P(Z > 0.236) = 1 - P(Z < 0.236) \\\\\[ P(Z > 0.236) = 1 - 0.5934 \\\\\[ P(Z > 0.236) \approx 0.4066 \][/tex]
Hello!! i’m not sure how to do this question, if you could explain your work that’d b great!!
[tex]\bf \sqrt{xy}=y\implies \left( xy \right)^{\frac{1}{2}}=y\implies \stackrel{\textit{chain rule~\hfill }}{\cfrac{1}{2}(xy)^{-\frac{1}{2}}\stackrel{\textit{product rule}}{\left(y+x\cfrac{dy}{dx} \right)}}=\cfrac{dy}{dx} \\\\\\ \cfrac{1}{2\sqrt{xy}}\left(y+x\cfrac{dy}{dx} \right)=\cfrac{dy}{dx}\implies \cfrac{y}{2\sqrt{xy}}+\cfrac{x}{2\sqrt{xy}}\cdot \cfrac{dy}{dx}=\cfrac{dy}{dx}[/tex]
[tex]\bf \cfrac{x}{2\sqrt{xy}}\cdot \cfrac{dy}{dx}=\cfrac{dy}{dx}-\cfrac{y}{2\sqrt{xy}} \implies \cfrac{x}{2\sqrt{xy}}\cdot \cfrac{dy}{dx}-\cfrac{dy}{dx}=-\cfrac{y}{2\sqrt{xy}} \\\\\\ \stackrel{\textit{common factor}}{\cfrac{dy}{dx}\left( \cfrac{x}{2\sqrt{xy}}-1 \right)}=-\cfrac{y}{2\sqrt{xy}} \implies \cfrac{dy}{dx}=-\cfrac{y}{\left( \frac{x}{2\sqrt{xy}}-1 \right)2\sqrt{xy}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \cfrac{dy}{dx}=-\cfrac{y}{x-2\sqrt{xy}}~\hfill[/tex]
Translate the phrase "" Nine times the difference of a number and 8"" into a algebraic expression . Simplify your result
click on picture, sorry if it's hard to read, but my phone messed up the typing
The phrase 'Nine times the difference of a number and 8' is translated into the algebraic expression 9(n - 8) and simplified to 9n - 72.
The phrase 'Nine times the difference of a number and 8' translates to an algebraic expression by following specific mathematical operations. To represent an unknown number, we use a variable, such as 'n', and the phrase 'the difference of a number and 8' would be written as 'n - 8'. To adhere to the phrase 'nine times', we multiply the difference by 9, leading to the expression 9(n - 8).
When we simplify the expression, we need to distribute the 9 to both terms within the parentheses: 9 × n and 9 × (-8), which gives us 9n - 72. Thus, the simplified algebraic expression for the phrase 'Nine times the difference of a number and 8' is 9n - 72.
Find the angle 0 between the vectors. u=(1, 1, 1, 0), v = (4, 4, 4, 4).
Answer:
30 degrees
Step-by-step explanation:
u dot v=1*4+1*4+1*4+0*4=4+4+4+0=12
|u|=sqrt(1^2+1^2+1^2+0^2)=sqrt(3)
|v|=sqrt(4^2+4^2+4^2+4^2)=sqrt(4*4^2)=2*4=8
cos(theta)=u dot v/(|u||v|)
cos(theta)=12/(sqrt(3)*8)
cos(theta)=3/(sqrt(3)*2)
cos(theta)=sqrt(3)/2
theta=30 degrees
A rectangular aquarium has length (x+ 10), width (x + 4), and height (t + 6). Determine a simplified function that represents the volume of the aquarium. [2 Marks)
Answer:
V = x³ + 20x² + 124x + 240
Step-by-step explanation:
Volume of a rectangular prism is width times length times height.
V = wlh
Given w = x+4, l = x+10, and h = x + 6:
V = (x + 4)(x + 10)(x + 6)
V = (x + 4)(x² + 16x + 60)
V = x²(x + 4) + 16x(x + 4) + 60(x + 4)
V = x³ + 4x² + 16x² + 64x + 60x + 240
V = x³ + 20x² + 124x + 240
Final answer:
The volume of the rectangular aquarium is given by the function V = x²t + 6x² + 14xt + 84x + 40t + 240, representing the product of its length, width, and height with given dimensions.
Explanation:
To determine a simplified function that represents the volume of the aquarium with given dimensions, we need to use the formula for the volume of a rectangular prism, which is length × width × height. The problem provides expressions for these dimensions: length is (x + 10), width is (x + 4), and height is (t + 6).
Therefore, the volume V of the aquarium can be calculated as follows:
V = (x + 10) × (x + 4) × (t + 6)
To simplify this, we multiply the expressions:
V = (x² + 14x + 40)(t + 6)
Expanding this, we get:
V = x²t + 6x² + 14xt + 84x + 40t + 240
This is the simplified function for the volume of the aquarium in terms of x and t.
Choose the property used to rewrite the expression. log base 4, 7 + log base 4, 2 = log base 4, 14
Answer:
[tex] log_{a}(x) + log_{a}(y) = log_{a}(xy) [/tex]
In this high school level mathematics problem, the Product Rule of Logarithms is applied to rewrite the given expression using the appropriate property.
The property used to rewrite the expression is the Product Rule of Logarithms. According to this property, when adding two logarithms with the same base, it is equivalent to multiplying the values inside the logarithms.
So, log base 4 of 7 + log base 4 of 2 can be rewritten as log base 4 of (7*2), which simplifies to log base 4 of 14.
A class of 32 students is organised in 33 teams every team consists of 3 students and there are no identical teams . show that there are two teams with exactly one common student
Answer:
Step-by-step explanation:
Let's start by making up as many teams as we can with the 32 student. Given that each team is different, we can make 10 teams of 3 each. (we still have 23 more teams to make).
The last two people make a team of only 2. No matter which student from the 30 other students is picked, the team of two and the one the student is coming from will have one student in common. Though there are more borrowings that take place (many more), the results remain as stated. At least 2 teams will have 1 person in common.
The method is called the pigeon hole method.
By applying the Pigeonhole Principle in combinatorics, in a scenario where 32 students are assigned to 33 teams of 3 students each, there must exist two teams that share exactly one student.
Explanation:This problem can be solved by using the principles of Combinatorics and the Pigeonhole Principle. The Pigeonhole Principle states that if you try to distribute n items into m containers and n > m, then at least one container must contain more than one item.
In the given scenario, we have 32 students that are being assigned to 33 teams, with each team consisting of 3 students. That means a total of 96 (3 x 32) places in teams.
If each student is a 'pigeon' and each 'place' in a team is a 'pigeonhole', the Pigeonhole Principle tells us that at least two pigeons must share at least one pigeonhole. Since each student can't be in more than one place at a time nor in the same team more than once, there must exist two teams that share exactly one student.
Learn more about Pigeonhole Principle here:https://brainly.com/question/34617354
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