Step-by-step explanation:
First, use trig to find the height of the slide.
The slide forms a right triangle. We know the hypotenuse is 8.80 ft, and the angle opposite of the height is 25.0°. So using sine:
sin 25.0° = h / 8.80
h = 3.72 ft
Converting to meters:
h = 3.72 ft × (1 m / 3.28 ft)
h = 1.13 m
Potential gravitational energy is:
PE = mgh
where m is the mass, g is the acceleration due to gravity, and h is the relative height.
At the bottom of the slide, h = 0:
PE = (63.0 kg) (9.8 m/s²) (0 m)
PE = 0 J
At the top of the slide, h = 1.13 m:
PE = (63.0 kg) (9.8 m/s²) (1.13 m)
PE = 700 J
The change is the final potential energy minus the initial potential energy.
ΔPE = 0 J - 700 J
ΔPE = -700 J
In a class of 50 students, everyone has either a pierced nose or a pierced ear. The professor asks everyone with a pierced nose to raise his or her hand. Seven hands go up. Then the professor asked everyone with a pierced ear to do likewise. This time there are 46 hands raised. How many students have piercings both on their ears and their noses?
Answer: 3
Step-by-step explanation:
Let E be the event of that student pierces ear and N be the event of that student pierces nose.
Given: [tex]n(E\cup N=50)[/tex]
[tex]n(E)=46\\\\n(N)=7[/tex]
For any two event A and B, we have
[tex]n(A\cup B)=n(A)+n(B)-n(A\cap B)[/tex]
Similarly , [tex]n(E\cup N)=n(E)+n(N)-n(E\cap N)[/tex]
[tex]50=46+7-n(E\cap N)\\\\\Rightarrow\ n(E\cap N)=53-50=3[/tex]
Hence, 3 students have piercings both on their ears and their noses.
Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. 1) 30 and 40 A) No B) 37,39 C) 31, 33 D) 35, 37 2) 4 and 15 A) 5,7 and 9, 11 B) No C) 5,7 D) 5, 7 and 11, 13 3) 16 and 24 A) 17, 19 and 21, 23 C) 21,23 B) No D) 17, 19 4) 35 and 50 A) 41, 42 and 47,49 C) No B) 43, 47 D) 41, 43
Final answer:
Twin primes are pairs of prime numbers with a difference of two. Within the given ranges, the twin primes identified are (37, 39) between 30 and 40, (5, 7) and (11, 13) between 4 and 15, (17, 19) between 16 and 24, and (41, 43) between 35 and 50.
Explanation:
The subject of this question is identifying twin primes within a given range of numbers. Twin primes are pairs of prime numbers that have a difference of two. For example, (3, 5) and (11, 13) are twin primes because each pair consists of two prime numbers that are exactly two units apart. Let's identify the twin primes within the ranges provided:
Between 16 and 24: The twin primes are (17, 19), so the answer is D) 17, 19.
The mean length of six-year-old rainbow trout in the Arolik River in Alaska is 481 millimeters with a standard deviation of 41 millimeters. Assume these lengths are normally distributed. What proportion of six-year-old rainbow trout are less than 501 millimeters long?
Answer: 0.688
Step-by-step explanation:
Given: Mean : [tex]\mu = 481 \text{ millimeters}[/tex]
Standard deviation : [tex]\sigma=871\text{ millimeters}[/tex]
Sample size : [tex]n=1600[/tex]
We assume these lengths are normally distributed.
Then the formula to calculate the z score is given by :-
[tex]z=\dfrac{X-\mu}{\sigma}[/tex]
For X=501
[tex]z=\dfrac{501-481 }{41}=0.487804878049\approx0.49[/tex]
The p-value of z =[tex]P(z<0.49)=0.6879331\approx0.688[/tex]
Now, the probability of the newborns weighed between 1492 grams and 4976 grams is given by :-
Hence, The proportion of six-year-old rainbow trout are less than 501 millimeters long = 0.688
Final answer:
To determine the proportion of six-year-old rainbow trout less than 501 millimeters long, we calculate the z-score and then find the corresponding proportion in the standard normal distribution.
Explanation:
The question asks for the proportion of six-year-old rainbow trout in the Arolik River in Alaska that are less than 501 millimeters long, given a mean length of 481 millimeters and a standard deviation of 41 millimeters, with lengths normally distributed. To find this proportion, we need to calculate the z-score for 501 millimeters.
First, we calculate the z-score using the formula:
z = (X - μ) / σ
where X is the value we are evaluating (501 millimeters), μ is the mean (481 millimeters), and σ is the standard deviation (41 millimeters).
Plugging the values in, we get:
z = (501 - 481) / 41
z = 20 / 41
z ≈ 0.49
Next, we consult the standard normal distribution table or use statistical software to find the proportion of values below a z-score of 0.49.
This will give us the proportion of six-year-old rainbow trout that are less than 501 millimeters long.
A pair of fair dice is tossed. Events A and B are defined as follows. A: {The sum of the numbers on the dice is 4} B: {The sum of the numbers on the dice is 11} Identify the sample points in the event A ∪ B. There are no sample points in the event A ∪ B. {(1, 4), (2, 3), (3, 2), (4, 1), (5, 6), (6, 5)} {(1, 3), (2, 2), (3, 1), (5, 6), (6, 5)} {(1, 4), (2, 2), (4, 1), (5, 6), (6, 5)}
Answer: 5/36
Step-by-step explanation: We know that A U B means all the possible combinations that make event A or event B true. As we know, the only combinations that can make 4 is 1+3, 2+2, or 3+1, and the only combinations that can maker 11 is 5+6 and 6+5. This leaves us with a total of 5 combinations, and with a total of 36 combinations, that means that there is a 5/36 chance that the combinations of the dice add to either 4 or 11.
Final answer:
The sample points in event A ∪ B are (1, 4), (2, 3), (3, 2), (4, 1), (5, 6), (6, 5).
Explanation:
The sample points in the event A ∪ B are the points that belong to either event A or event B. To find the sample points in A ∪ B, we combine the sample points from A and B.
The sample points in event A are (1, 4), (2, 3), (3, 2), (4, 1), (5, 6), (6, 5) and the sample points in event B are (2, 3), (5, 6), (6, 5). When we combine these sample points, we have the following sample points in A ∪ B: (1, 4), (2, 3), (3, 2), (4, 1), (5, 6), (6, 5).
The investigating team suspected that there were differences in the cost of repairing cars in workshop I and workshop II. The investigating team suspected that the costs raised by workshop I were greater than workshop II. For that they tested the repair of 15 cars in each workshop to see the cost of repairs.The decision of the right hypothesis to prove the suspicion above is a. H0 : μ1- μ2 = 0; HA : μ1- μ2 ≠ 0
b. H0 : μ1- μ2 ≥ 0; HA : μ1- μ2 < 0
c. H0 : μD ≥ 0; HA : μD < 0 with μD = μ2- μ1
d. H0 : μD = 0; HA : μD ≠ 0 with μD = μ2- μ1
e. H0 : μ1- μ2 = 0; HA : μ1- μ2 ≥ 0
Answer:
[tex]H0 : \muD = 0\\\\ H_A : \mu D \neq0 \text{ with }\ \mu D = \mu2- \mu1[/tex]
Step-by-step explanation:
Let [tex]\mu_1[/tex] and [tex]\mu_2[/tex] are the mean costs raised by workshop I and workshop II respectively.
Given claim :The costs raised by workshop I were greater than workshop II.
i.e. [tex]\mu_1>\mu_2\ or\ \mu_1-\mu_2>0[/tex]
Since it does not contain equals sign therefore we consider it as the alternative hypothesis.
The null hypothesis will be just opposite of this.
i.e. [tex]H_0:\mu_1-\mu_2\leq0[/tex]
Hence, The decision of the right hypothesis to prove the given suspicion:-
[tex]H_0:\mu_1-\mu_2\leq0\\\\H_1:\mu_1-\mu_2>0[/tex]
[tex]\text{If }\ D=\mu_1-\mu_2[/tex], then
[tex]H0 : \mu D = 0\\\\ H_A : \mu D \neq0 \text{ with }\ \mu D = \mu_2- \mu_1[/tex]
What are the solutions of the equation x4 – 5x2 – 14 = 0? Use factoring to solve.
Someone help please!!!
Answer: x=√7
x=√2i
We'll use factoring by grouping to solve the equation. This method involves grouping the terms of the polynomial into two binomials, such that the product of the leading coefficients of the binomials is equal to the constant term, and the sum of the products of the remaining terms is equal to the middle term.Steps to solve:
1. Factor the expression:
(x²−7)(x²+2)=0. Create separate equations and solve:
x²−7=0
x²+2=0. Solve the first equation:
x²−7=0
x=±√7. Solve the second equation:
x²+2=0
x=±√2i
A dot-com company plans to place money in a new venture capital fund that currently returns 18% per year, compounded daily, what effective rate is this (a) yearly and (b) semi-annually?
Answer:it is 18*20 effective every year
Step-by-step explanation:
To find the effective annual rate for an 18% yearly interest compounded daily, the formula (1 + i/n)^(n*t) - 1 is used.
For a yearly rate, 'n' equals 365 and 't' equals 1, while for a semi-annual rate, 't' is 0.5. These calculations show the true interest earnings per period.
Explanation:To calculate the effective annual rate (EAR) for an 18% per year, compounded daily interest rate, we use the formula for the EAR, which is:
EAR = (1 + i/n)^(n*t) - 1
Where i is the nominal interest rate, n is the number of compounding periods per year, and t is the number of years. For daily compounding, n = 365, as there are 365 days in a year.
The calculation for a yearly (annual) effective rate would be:
EAR = (1 + 0.18/365)^(365*1) - 1
For a semi-annual effective rate, we consider the compounding effect over half a year, so t = 0.5:
EAR semi-annual = (1 + 0.18/365)^(365*0.5) - 1
These calculations give you the effective rate of interest for each compounding period.
The mean number of flight hours for Continental Airline pilots is 49 hours per month. Assume that this mean was based on a sample of 100 Continental pilots and that the sample standard deviation was 11.5 hours. (a) Calculate the margin of error for a 95% confidence interval. (b) Calculate the upper bound for a 95% confidence interval.
Answer with explanation:
(a)
Mean number of flight hours for Continental Airline pilots = 49 hours per month
Total Sample Size =100
Standard Deviation =11.5 Hours
Margin of error for a 95% confidence interval
[tex]=Z_{95 \text{Percent}}\times \frac{\sigma}{\sqrt{n}}\\\\=0.8365 \times \frac{11.5}{\sqrt{100}}\\\\=\frac{9.61975}{10}\\\\=0.961975\\\\=0.97(\text{Approx})[/tex]
(b)
The Range of values for a 95% confidence interval
⇒ Mean number of flight + Margin of Error ≤ Confidence interval ≤ Mean number of flight - Margin of Error
⇒ 49+0.97 ≤ Confidence interval ≤ 49-0.97
⇒ 49.97 ≤ Confidence interval ≤48.03
Upper Bound = 49.97
Suppose that on January 1, 2018, you buy a bond for $2,000 that will pay interest of 3.6% per year compounded continuously for 20 years. You never withdraw any of the interest earned on the bond. (a) What will the bond be worth on January 1, 2038?
Answer: $4108.87
Step-by-step explanation:
Given : Present value : [tex]P= \$2,000[/tex]
The number of time period : [tex]t=20\text{ years}[/tex]
The rate of interest : [tex]r=3.6\%\ =0.036[/tex]
Let P be the present value of bond .
The formula to calculate the future value is given by :-
[tex]FV=Pe^{rt}[/tex]
[tex]\\\\\Rightarrow\ FV=2000e^{0.036\times20}}\\\\\Rightarrow\ FV=4108.86642129\approx4108.87[/tex]
Hence, the future value of the bond on January 1, 2038 would be $4108.87 .
The date of death for a widow was 2017. If the estate was valued at $7.36 million and the estate was taxed at 40 percent, what was the heir's tax liability? (Enter your answer in dollars not in millions.) Heir's tax liability
Answer:
zero
Step-by-step explanation:
The inheritance tax is paid by the estate. The heirs have no tax liability on the amount inherited.
Final answer:
The heir's tax liability for an estate valued at $7.36 million, after applying a 40% tax rate to the amount above the estate tax exemption of $5.43 million (assumed for 2017), would be $772,000.
Explanation:
To calculate the heir's tax liability for an estate valued at $7.36 million with a tax rate of 40%, we need to determine if the estate's value exceeds the estate tax exemption threshold. According to the Center on Budget and Policy Priorities, in 2015, the exemption limit was $5.43 million. Since the date of death is 2017, the exemption amount may have been different, but for this calculation, we’ll assume it is the same.
Here's how to calculate the tax liability:
Subtract the exemption limit from the total estate value: $7,360,000 - $5,430,000 = $1,930,000. This is the taxable amount.
Multiply the taxable amount by the tax rate: $1,930,000 * 40% = $772,000.
Therefore, the heir's tax liability would be $772,000.
The number of accidents at a dangerous intersection in Smalltown during each of the last six years is as follows: 0, 1, 1, 2, 3, 5. For this data set, the standard deviation of the number of accidents in a year (rounded to the nearest tenth) is: (a) Mean < Median < Mode (b) Mean < Mode < Median (c) Median < Mean < Mode (d) Mode < Mean < Median (e) Mode < Median < Mean
Answer: Option 'e' is correct.
Step-by-step explanation:
Since we have given that
The number of accidents in Small town during the last six years as follows:
0,1,1,2,3,5.
First we calculate :
1) Mean :
[tex]\bar{X}=\dfrac{0+1+1+2+3+5}{6}=\dfrac{12}{6}=2[/tex]
2) Median:
0,1,1,2,3,5
As we know that "Median" is the middle value of data:
Median = [tex]\dfrac{1+2}{2}=\dfrac{3}{2}=1.5[/tex]
3) Mode:
0,1,1,2,3,5
As we know that Mode is the most occurring element among the data.
So, Mode = 1
Now, we can say that Mode< Median < Mean
Hence, Option 'e' is correct.
Answer:
Correct answer is (E)
Step-by-step explanation:
Took the test on Plato Math and got it right
Hope I helped :D
Problem #1. (a) Prove that if z is irrational, then underroot x is irrational.
Answer with explanation:
It is given that z is an irrational complex number.
Z= x + i y
Where x is real part and y is Imaginary part.x and y can be any Real number.
If z is an irrational complex number , then both real part and imaginary part should be a complex number.That is x, y ∈Q, then ,Q= Set of Irrationals.
It means , x and y both should be an irrational number.
So, if x is any irrational number then, [tex]\sqrt{x}[/tex] will be also an irrational number.
13. Justin generously lends $500,000 to his friend, Jamie. However, being a maleficent businessman, he charges Jamie interest at 8.6 % per annum compounded quarterly. How much will Jamie owe Justin after 25 years? What is the accumulated interest? [3 marks]
Answer:
Jamie will owe $ 41,95,725.83 ( approx ),
Accumulated interest is $ 36,95,725.83
Step-by-step explanation:
Since, the amount formula in compound interest is,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where, P is the principal amount,
r is the annual rate of interest,
n is the compounding periods in a year,
t is the time in years,
Here, P = $ 500,000,
r = 8.6 %=0.086,
n = 4,
t = 25 years,
By substituting the values,
[tex]A=500000(1+\frac{0.086}{4})^{100}[/tex]
[tex]=500000(1+0.0215)^{100}[/tex]
[tex]=500000(1.0215)^{100}[/tex]
[tex]=4195725.82746[/tex]
[tex]\approx 4195725.83[/tex]
Also, the accumulated interest = A-P = 4195725.83 - 500000 = $ 3695725.83
Given the following statistics for women over the age of 50 entering our medical clinic: 1% actually have breast cancer 90% of the women who have breast cancer are going to get a positive test result (affirming that they have the disease) 8% of those that actually don’t have the disease are going to be told that they do have breast cancer (a “false positive”) What’s the actual probability, if a woman gets a positive test result, that she actually does have breast cancer?
Answer: 91.2%
Step-by-step explanation:
Of the 1% of women that have breast cancer, 90% of those are tested positive with 8% of those being false positive. 8% of 90% is 8.8% so 91.2% chance.
need help with probability questions
[tex]|\Omega|=5^3=125\\|A|=2\cdot3\cdot1=6\\\\P(A)=\dfrac{6}{125}=\dfrac{48}{1000}=0.048[/tex]
An experiment requires a sequence of three steps. The first step can result in four possible outcomes, the second in three possible outcomes, and the third in two possible outcomes. What is the total number of outcomes possible? HINT [See Quick Example on page 419.]
The total number of possible outcomes of the experiment, where the first step has four possible outcomes, the second has three, and the third has two, is 24. This is calculated by multiplying the number of outcomes for each step together.
Explanation:This question is based on the concept of combinations in mathematics, specifically when dealing with the outcomes of sequences or actions. Each step in a sequence can have several possible outcomes, and each combination of actions from each step is considered a unique sequence. Because the steps are independent, the total number of possible outcomes is the product of the number of possible outcomes for each step.
In this particular example: the first step has four possible outcomes, the second step has three possible outcomes, and the third step has two possible outcomes. Therefore, to find the total number of outcomes feasible, you simply need to multiply these numbers together:
4 (outcomes from step one) × 3 (outcomes from step two) × 2 (outcomes from step three) = 24 possible outcomes in total.
Learn more about Combinations here:https://brainly.com/question/39347572
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If sinθ = -1/2 and θ is in Quadrant III, then tanθ
let's recall that on the III Quadrant sine/y is negative and cosine/y is negative, now, the hypotenuse/radius is never negative, since it's just a radius unit.
[tex]\bf sin(\theta )=\cfrac{\stackrel{opposite}{-1}}{\stackrel{hypotenuse}{2}}\impliedby \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2-b^2}=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm\sqrt{2^2-(-1)^2}=a\implies \pm\sqrt{4-1}=a\implies \pm\sqrt{3}=a\implies \stackrel{\textit{III Quadrant}}{-\sqrt{3}=a} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf tan(\theta )=\cfrac{\stackrel{opposite}{-1}}{\stackrel{adjacent}{-\sqrt{3}}}\implies \stackrel{\textit{rationalizing the denominator}}{tan(\theta )=\cfrac{-1}{-\sqrt{3}}\cdot \cfrac{\sqrt{3}}{\sqrt{3}} }\implies tan(\theta )=\cfrac{\sqrt{3}}{3}[/tex]
PLEASE HELP ME
Tony bought a desktop computer and a laptop computer. Before finance charges, the laptop cost $450 more than the desktop. He paid for the computers using two different financing plans. For the desktop the interest rate was 9% per year, and for the laptop it was 6 % per year. The total finance charges for one year were $300
. How much did each computer cost before finance charges?
Answer:
Laptop: $2,270
Desktop: $1,820
Step-by-step explanation:
Let L identify the laptop price and D the desktop price.
We can first say:
L = D + 450 ( the laptop cost $450 more than the desktop)
Then we can say:
0.09 D + 0.06 L = 300 (The total finance charges for one year were $300)
Then we substitute L by its value from first equation into the second equation:
0.09 D + 0.06 (D + 450) = 300
0.09 D + 0.06 D + 27 = 300
0.15D = 273 (removed 27 on both sides, and simplified left side)
D = 1,820
The cost of the desktop was $1,820
The cost of the laptop was $2,270 (price of desktop + $450)
Final answer:
By setting up equations based on the given finance charges and interest rates, we find that before finance charges, the desktop computer cost $1820 and the laptop cost $2270.
Explanation:
The student's question asks to determine the cost of each computer before finance charges. Let's denote the cost of the desktop computer as D and the cost of the laptop as L. From the information provided, we know that L = D + $450. The total finance charges for the desktop at 9% per year and for the laptop at 6% per year amount to $300. Hence, the equation for the finance charge can be written as 0.09D + 0.06L = $300. Substituting the expression for L from the first equation into the second, we get 0.09D + 0.06(D + $450) = $300, which simplifies to 0.09D + 0.06D + $27 = $300. Adding the D terms together, we get 0.15D + $27 = $300. Subtracting $27 from both sides, we obtain 0.15D = $273. Dividing both sides by 0.15, the cost of the desktop computer is found to be D = $1820. To find the cost of the laptop, we use the first equation: L = $1820 + $450 = $2270.
In conclusion, before finance charges, the desktop computer cost $1820 and the laptop $2270.
The box plots show the data distributions for the number of customers who used a coupon each hour for two days of a store sale.
What is the difference of the medians?
Answer:
2
Step-by-step explanation:
We can observe from the box plot the medians of both days.
The line in the middle of the box plot represents the median.
The median for Day 1 is: 6
The median for Day 2 is: 8
We have to find the difference between medians of both box plots so the difference is:
8 - 6 = 2
The difference between the medians is 2 ..
Answer:
its B
Step-by-step explanation:
ASAP: Which conclusion can be drawn based on the true statements shown?
If a triangle is equilateral, then all the sides in the triangle are congruent.
If all the sides in a triangle are congruent, then all the angles are congruent.
Based on the law of syllogism, if all the angles in a triangle are congruent, then the triangle is equilateral.
Based on the law of detachment, if all the angles in a triangle are congruent, then the triangle is equilateral.
Based on the law of syllogism, if a triangle is equilateral, then all the angles are congruent.
Based on the law of detachment, if a triangle is equilateral, then all the angles are congruent.
Answer:
Conclusion:
Based on the law of syllogism, if a triangle is equilateral, then all the angles are congruent.
Step-by-step explanation:
The law of syllogism, also called reasoning by transitivity, is a valid argument form of deductive reasoning that follows a set pattern. It is similar to the transitive property of equality, which reads: if a = b and b = c then, a = c.
In syllogism, we combine two or more logical statements into one logical statement.
Statement 1:
If a triangle is equilateral, then all the sides in the triangle are congruent.
Statement 2:
If all the sides in a triangle are congruent, then all the angles are congruent.
Concluded Statement using law of syllogism:
Based on the law of syllogism, if a triangle is equilateral, then all the angles are congruent.
Answer:
Based on the law of syllogism, if a triangle is equilateral, then all the angles are congruent.
Step-by-step explanation:
took test and made 100. law of syllogism shows this to be true
A manufacturer produces gears for use in an engine’s transmission that have a mean diameter of 10.00 mm and a standard deviation of 0.03 mm. The length of these diameters follows the normal distribution. What is the probability that a randomly selected gear has a diameter between 9.96 mm and 10.01 mm?
Answer: 0.2789
Step-by-step explanation:
Given: Mean : [tex]\mu=10.00\ mm [/tex]
Standard deviation : [tex]\sigma =0.03\ mm[/tex]
The formula to calculate z-score is given by :_
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 9.96 mm, we have
[tex]z=\dfrac{10-9.96}{0.03}\approx1.33[/tex]
For x= 10.01 mm, we have
[tex]z=\dfrac{10.01-10}{0.03}\approx0.33[/tex]
The P-value = [tex]P(0.33<z<1.33)=P(z<1.33)-P(z<0.33)[/tex]
[tex]= 0.9082408- 0.6293=0.2789408\approx0.2789[/tex]
Hence, the probability that a randomly selected gear has a diameter between 9.96 mm and 10.01 mm =0.2789
Answer:
Pr=0.2894
Step-by-step explanation:
given mean diameter =10 mm
standard deviation=0.03 mm
z equation is
z=x-μ/σ
The problem has two values of x
for x=9.96
z=-1.33
for x-10.01
z=0.33
from Probability table we have
Pr(-1.33<z<0.33)=pr(z<0.33)-pr(z>-1.33)
Pr=0.2894
if i have 18 days to complete assignments and i have 44 assignments to do, how many assignments do i have to do a day?
For 11 days, you'd do 3 assignments. That'll knock off 33 assignments. Then, for 5 days, you'll 2 assignments, which will leave you with 2 assignments. Then for one day, you'll only have to do 1 assignment. The last day you are free!!! :)
PLEASE HELP MEEEE!!!!!!
Answer:
g(x)
Step-by-step explanation:
-4/-4 = aops
aops = 1
f(1) = 5
f(x)'s max = 5
g(x)'s = 6
(pls give brainliest)
If f(x)=x^4+6, g(x)=x-2 and h(x)= sqrt (x), then f(g(h(x)))=
Please help ASAP I'm really confused with this math problem!!! :(
Answer:
x^2 + 4x * (3 - sqrt(x)) - 2(5 + sqrt(x))
Step-by-step explanation:
Firstly let us split this up, we need to first work out what g(h(x)) is:
h(x) = Sqrt(x) so g(h(x)) = g(sqrt(x)) = sqrt(x) - 2
Now to work out f(g(h(x))) = f(sqrt(x) - 2) = (sqrt(x) - 2)^4 + 6
= (sqrt(x) - 2) * (sqrt(x) - 2) * (sqrt(x) - 2) * (sqrt(x) - 2) - 6
= (x - 2 * sqrt(x) + 4) * (x - 2 * sqrt(x) + 4) - 6
= x^2 - 2x * sqrt(x) + 4x - 2x * sqrt(x) + 4x - 8 * sqrt(x) + 4x - 8 * sqrt(x) + 16 - 6
= x^2 - 4x * sqrt(x) + 12x - 16 * sqrt(x) + 10
= x^2 + 4x * (3 - sqrt(x)) - 2(5 + sqrt(x))
There are $528 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $9 per foot, and the fencing for the other three sides costs $3 per foot. The picture on the right depicts this situation. Consider the problem of finding the dimensions of the largest possible garden.
Answer:
22 ft by 44 ft, with 22 ft parallel to the road
Step-by-step explanation:
Problems in optimizing rectangular area for a given perimeter or perimeter cost all have a similar solution: the length (or cost) of one pair of opposite sides is equal to that of the other pair of opposite sides.
Here, that means that the sides perpendicular to the road will have a total cost of $528/2 = $264, so will have a total length of $264/($3/ft) = 88 ft. Since it is a rectangle, the dimension perpendicular to the road is 44 ft.
Likewise, the sides parallel to the road will have a total cost of $264. If x is the length in that direction, this means ...
9x +3x = 264
12x = 264
264/12 = x = 22
The length of the garden parallel to the road is 22 ft.
_____
If you solve this directly, you get the same result. Let x be the distance parallel to the road. Then the cost of fence for the two sides parallel to the road is (3x +9x) = 12x.
The length of fence perpendicular to the road will use the remaining cost, so that length will be (528 -12x)/(2·3). (Half of the remaining fence is used on each of the two parallel sides.) This expression for length simplifies to (88-2x).
Then the area of the garden will be the product of its length and width:
area = x(88 -2x)
This is the equation for a downward-opening parabola with zeros at x=0 and x=44. The vertex is located halfway between those zeros, at x = 22.
The dimensions of the largest garden are 22 ft parallel to the road and 44 ft perpendicular to the road.
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Write a quadratic function in standard form whose graph passes through (-5,0), (9,0), and (8, -39).
f(x) =
Answer:
[tex]y=3x^2-12x-135[/tex]
Step-by-step explanation:
The standard form of a quadratic is [tex]y=ax^2+bx+c[/tex]
We will use the x and y values from each of our 3 points to find a, b, and c. Filling in the x and y values from each point:
First point (-5, 0):
[tex]0=a(-5)^2+b(-5)+c[/tex] and
0 = 25a - 5b + c
Second point (9, 0):
[tex]0=a(9)^2+b(9)+c[/tex] and
0 = 81a + 9b + c
Third point (8, -39):
[tex]-39=a(8)^2+b(8)+c[/tex] and
-39 = 64a + 8b + c
Use the elimination method of solving systems on the first 2 equations to eliminate the c. Multiply the first equation by -1 to get:
-25a + 5b - c = 0
81a + 9b + c = 0
When the c's cancel out you're left with
56a + 14b = 0
Now use the second and third equations and elimination to get rid of the c's. Multiply the second equation by -1 to get:
-81a - 9b - c = 0
64a + 8b + c = -39
When the c's cancel out you're left with
-17a - 1b = -39
Between those 2 bolded equations, eliminate the b's. Do this by multiplying the second of the 2 by 14 to get:
56a + 14b = 0
-238a - 14b = -546
When the b's cancel out you're left with
-182a = -546 and
a = 3
Use this value of a to back substitute to find b:
56a + 14b = 0 so 56(3) + 14b = 0 gives you
168 + 14b = 0 and 14b = -168 so
b = -12
Now back sub in a and b to find c:
0 = 25a - 5b + c gives you
0 = 75+ 60 + c so
0 = 135 + c and
c = -135
Put that all together into the standard form equation to get
[tex]y=3x^2-12x-135[/tex]
Answer:
f(x) = 3x^2 -12x -135
Step-by-step explanation:
The given zeros tell you that two factors are (x +5) and (x -9). Then the function can be written ...
f(x) = a(x +5)(x -9)
We can find "a" from ...
f(8) = -39 = a(8 +5)(8 -9) = -13a
3 = a . . . . . . divide by -13
Expanding the above form, we get the standard form ...
f(x) = 3x^2 -12x -135
In a certain normal distribution of scores, the mean is 30 and the standard deviation is 3. Find the z-score corresponding to a score of 23.
Answer:
-2.333
Step-by-step explanation:
In a certain normal distribution of scores, the mean is 30 and the standard deviation is 3. Find the z-score corresponding to a score of 23.
The z-score corresponding to an observed value in a normal distribution is calculated as;
z-score = (observed value - mean)/(standard deviation)
Our observed score is 23, the mean is 30, and the standard deviation is 3. The z-score will thus be;
z-score = ( 23 - 30)/( 3)
z-score = -2.333
Answer:
- 2.333
Step-by-step explanation:
The formulae to apply here is;
z= (x-μ) / δ------------where x is the score, μ is the mean and δ is the standard deviation
Given x=23, μ=30 and δ= 3
z= (23-30) / 3 z= - 7/3 z= -2.333
5) The mean salary of 5 employees is $40300. The median is $38500. The lowest paid employee's salary is $32000. If the lowest paid employee gets a $3100 raise, then ...
a) What is the new mean?
New Mean = $
b) What is the new median?
New Median = $
Answer:
New mean = $40920
New Median = $38500 (Same as old)
Step-by-step explanation:
Given
Mean = $40300
Median = $38500
The formula for mean is:
Mean = Sum/No. of Values
According to the formula,
40300 = Sum/5
40300*5 = Sum
Sum of Salaries = 201500
If the salary of lowest employee is raised 3100, then sum will be 204600.
The new mean will be:
Mean = 204600/5
Mean =$40920
The median will remain unchanged as it is the middle value of 5 quantities. 2 values are greater than median and two are less than median. Even after the raise in salary, the salary becomes $35100 which is still less than median. So the median will be same as old ..
With the raise, the new mean salary of the employees is $40920, while the median salary is likely to remain $38500 unless the raise causes a rearrangement in the salary sequence.
Explanation:Assuming there have been no other changes to the employees' salaries, we can calculate the new mean and new median as follows:
The original total salary of the five employees is $40300 times 5, which equals $201500. The lowest paid employee receives a $3100 raise, so the new total salary for the five employees is $201500 + $3100 = $204600. Now, to find the new mean, we divide the total by the number of employees, that is $204600 / 5 = $40920. So the new mean is $40920.The median salary is the middle value when salaries are arranged in order. Given we only have information about the lowest salary, not the distribution of the other salaries, it's impossible to calculate a new median with certainty. However, unless the increase causes the lowest salary ($32000 + $3100 = $35100) to exceed the next highest salary, the median would remain the same at $38500.Learn more about mean and median here:https://brainly.com/question/16083886
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How can you decompose the composite figure to determine its area?
A) As a circle, three rectangles, and a triangle
B) As a circle, a trapezoid, and four triangles
C) As a semicircle, three rectangles, and a square
D) As a semicircle, a trapezoid, and two rectangles
Answer: D) As a semicircle, a trapezoid, and two rectangles.
Step-by-step explanation:
In order to find the area of the composite figure provided, it is necessary to descompose it.
Observe the image attached.
You can observe that it can descomposed as:
1- A semicircle, whose area can be calculated with this formula:
[tex]A=\frac{\pi r^2}{2}[/tex]
Where r is the radius.
2- A trapezoid, whose area can be calculated with this formula:
[tex]A=\frac{h}{2}(B+b)[/tex]
Where h is the height, B is the larger base and b is the minor base.
3- Rectangle.
4- Rectangle.
The formula for calculate the area of a rectangle is:
[tex]A=lw[/tex]
Where l is the lenght and w is the width.
The area of the composite figure consists of a semicircle, a trapezoid, and two rectangles
What is a trapezoid?The Trapezoid is a 4 sided polygon. Two sides of the shape are parallel to each other and they are termed as the bases of the trapezoid. The non-parallel sides are known as the legs or lateral sides of a trapezoid.
There are three types of trapezoids , and those are given below:
a) Isosceles Trapezoid
b) Scalene Trapezoid
c) Right Trapezoid
The area of the Trapezoid is given by
Area of Trapezoid = ( ( a + b ) h ) / 2
where , a = shorter base of trapezium
b = longer base of trapezium
h = height of trapezium
Given data ,
The figure consists of a semicircle, a trapezoid, and two rectangles
So , the area of semicircle C = πr² / 2
The area of trapezoid T = ( ( a + b ) h ) / 2
And , the area of 2 rectangles R = 2 x L x B
Hence , the area of composite figure A = πr² / 2 + ( ( a + b ) h ) / 2 + 2 x L x B
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According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by 62 % to $470 thousand. What was the average price of a new home in 2000h The average price of a new home in 2000 was $ (Do not round until the final answer. Then round to the nearest thousand as needed)
Answer: $ 290 thousand
Step-by-step explanation:
Given : According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by 62 % to $470 thousand.
Let X be the the average price of a new home in 2000 .
Then , the 62 % increase in price is given by :-
[tex]x+0.62(x)=x(1+0.62)=1.62x[/tex]
Since , the the average price of the home in 2016 = $470 thosand
[tex]1.62x=470\\\\\Rightarrow\ x=\dfrac{470}{1.62}=290.12345679\approx290[/tex]
Hence, the average price of a new home in 2000 = $ 290 thousand .
The average price of a new house in 2000 is approximately 290,124 dollars.
What is the percentage?The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.
According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by 62 % to $470 thousand.
Let x be the average price of a new house in 2000. Then we have
[tex]\rm x = \dfrac{Present \ price}{1+Increased\ rate}\\\\\\x = \dfrac{470000}{1+0.62}\\\\\\x = \dfrac{470000}{1.62}\\\\\\x = 290123.4568 \approx 290124[/tex]
Thus, the average price of a new house in 2000 is 290,124 dollars.
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