Answer:
true its just a filp with the y and x and resolved
Step-by-step explanation:
Any help would be greatly appreciated
Step-by-step explanation:
4y + 6 = 16
you can find y (meters)
then replace y value in every side of this shape
then plus all line you will receive the perimeter
How much will a person pay for 11.6 pounds of bananas at a price of 1.51 per pound?
Answer:
17.52
Step-by-step explanation:
literally 11.6 x 1.51 and then round to hundredth place
What’s the correct answer for this?
Answer:
x = 19
Step-by-step explanation:
In the attached file
A group of hikers buy 8 bags of trail mix. Each bag contains 3 1/2 cups of trail mix. The trail mix is shared evenly among 12 hikers. How many cups of trail mix will each hiker receive?
Each hiker will receive cups of trail mix.
Answer:
Each hiker will get 2 1/3 cups
Step-by-step explanation:
8 bags multiplied by 3.5
28 cups overall
The 28 cups is shared between 12 hikers
28/12=2 1/3 cups
What’s the correct answer for this question?
Answer:
0.37
Step-by-step explanation:
As shown in table,
The probability of the fruit is orange is P(A) = 0.3
The probability of the fruit is organic is P(B)
The probability of the fruit is orange and organic is P(A⋂B) = 0.11
=> The probability that a randomly selected orange is organic is calculated by applying the conditional probability formula:
P(B|A) =P(A⋂B)/P(A) = 0.11/0.3 = 0.37
=> Option D is correct
Hope this helps!
The owner of Limp Pines Resort wanted to know the average age of its clients. A random sample of 25 tourists is taken. It shows a mean age of 46 years with a standard deviation of 5 years. The width of a 95 percent CI for the true mean client age is approximately: _________
Given Information:
Sample size = n = 25
Mean age of client = 46 years
Standard deviation of age of client = 5 years
Confidence level = 95%
Required Information:
Width of the confidence interval = ?
Answer:
[tex]$ \text {width of CI } = \pm 2.064 $\\\\[/tex]
Step-by-step explanation:
The width for the true mean client age is given by
[tex]$ \text {width of CI } =\pm t_{\alpha/2}(\frac{s}{\sqrt{n} } ) $[/tex]
Where n is the sample size, s is the standard deviation of age of client, and [tex]t_{\alpha/2}[/tex] is the t-score corresponding to 95% confidence level.
The t-score corresponding to 95% confidence level is
Significance level = 1 - 0.95 = 0.05/2 = 0.025
Degree of freedom = n - 1 = 25 - 1 = 24
From the t-table at α = 0.025 and DF = 24
t-score = 2.064
[tex]$ \text {width of CI } = \pm (2.064)(\frac{5}{\sqrt{25} } ) $\\\\[/tex]
[tex]$ \text {width of CI } =\pm (2.064)(\frac{5}{5 } ) $\\\\[/tex]
[tex]$ \text {width of CI } = \pm (2.064)(1) $\\\\[/tex]
[tex]$ \text {width of CI } = \pm 2.064 $\\\\[/tex]
Therefore, width of the 95% confidence Interval for the true mean client age is approximately ±2.064.
Bonus:
The corresponding 95% confidence interval is given by
[tex]$ \text {Confidence Interval } = \bar{x} \pm t_{\alpha/2}(\frac{s}{\sqrt{n} } ) $[/tex]
Where x_bar is the mean client age
[tex]$ \text {Confidence Interval } = 46 \pm 2.064 $[/tex]
[tex]$ \text {Confidence Interval } = (43.936, 48.064) $[/tex]
[tex]$ \text {Confidence Interval } = (44, 48) $[/tex]
Which means that we are 95% sure that the true mean client age is within the range of (44, 48) years.
Emma and Max want to buy a house together.
Emma earns £18,500 and Max earns £22,500. They have £6000 savings.
They want to buy a house that is being sold for £130,000.
They will pay the deposit with their savings and take out a mortgage to pay for the rest.
Emma and Max can borrow 3 times their combined incomes as a mortgage.
They will need to pay 5% of the selling price of the value as a deposit.
How much can they borrow as a mortgage?
Answer:
£123,000
Step-by-step explanation:
Their combined income is ...
£18,500 +22,500 = £41,000
They can borrow 3 times this amount, or ...
3 × £41,000 = £123,000
__
Comment on this transaction
The required deposit is 5% of 130,000 = 6,500, which is more than their savings. After this down payment is made, the remaining value is £123,500, which exceeds their borrowing power. It appears that Emma and Max need to find a house with a lower price.
Simplify the given expression. Cite a property from Theorem 6.2.2 for each step. (A − (A ∩ B)) ∩ (B − (A ∩ B)) Let A and B be any sets. Then (A − (A ∩ B)) ∩ (B − (A ∩ B)) = = (A ∩ (A ∩ B)c) ∩ (B ∩ (A ∩ B)c) = A ∩ ((A ∩ B)c ∩ (B ∩ (A ∩ B)c)) = A ∩ (((A ∩ B)c ∩ B) ∩ (A ∩ B)c) = A ∩ ((B ∩ (A ∩ B)c) ∩ (A ∩ B)c) = A ∩ (B ∩ ((A ∩ B)c ∩ (A ∩ B)c)) = A ∩ (B ∩ (A ∩ B)c) = (A ∩ B) ∩ (A ∩ B)c = ∅
Answer:
Step-by-step explanation:
Consider the sets A and B
(A − (A ∩ B)) ∩ (B − (A ∩ B))
= (A ∩ (A ∩ B)c) ∩ (B ∩ (A ∩ B)c) by the set difference law
= (A ∩ (Ac ∩ B)c) ∩ (B ∩ (Ac ∩ B)c) by De Morgan's law
= {(A ∩ Ac) ∪ (A ∩ Bc)} ∩ {(B ∩ Ac) ∪ (B ∩ Bc)} by the distributive law
= {∅ ∪ (A ∩ Bc)} ∩ {(B ∩ Ac) ∪ ∅} by complementation
= {A ∩ Bc} ∩ {B ∩ Ac} by identity law
= (A ∩ Ac) ∩ (B ∩ Ac) by the associative law
= ∅ ∩ ∅ by complementation
= ∅ by the universal bound law
Therefore, (A − (A ∩ B)) ∩ (B − (A ∩ B)) = ∅
Answer:
Considere los conjuntos A y B
(A − (A ∩ B)) ∩ (B − (A ∩ B))
= (A ∩ (A ∩ B)c) ∩ (B ∩ (A ∩ B)c) por la ley de diferencia establecida
= (A ∩ (Ac ∩ B)c) ∩ (B ∩ (Ac ∩ B)c) por la ley de De Morgan
= {(A ∩ Ac) ∪ (A ∩ Bc)} ∩ {(B ∩ Ac) ∪ (B ∩ Bc)} por la ley distributiva
= {∅ ∪ (A ∩ Bc)} ∩ {(B ∩ Ac) ∪ ∅} complementando
= {A ∩ Bc} ∩ {B ∩ Ac} por ley de identidad
= (A ∩ Ac) ∩ (B ∩ Ac) por la ley asociativa
= ∅ ∩ ∅ complementando
= ∅ por la ley universal consolidada
Step-by-step explanation:
PLEEEASE HELPPPPPPP!! substitution
Answer:
X=80
Y=9
Z=7
Step-by-step explanation:
What’s the correct answer for this?
Answer:
54
Step-by-step explanation:
Since diameter AB divides MN into two equal parts hence
MO = NO
NOW,
5x+34 = -2(1-7x)
5x+34 = -2+14x
34+2 = 14x-5x
36 = 9x
Dividing both sides by 9
x = 4
Now,
NO = -2(1-7(4))
NO = -2+56
NO = -2+56
NO = 54
How to do arithmetic sequence with fractions I’m so confused
Answer:
Step-by-step explanation:
first you have to see itis an arithmetic or geometric sequence.
then find common difference(for A.P) or common ratio (for G.P)
c.d .=an-an-1
c.r. =an/an-1
1.G.P.
2. A.P
3.G.P
3,5,25/3,125/9,...
c.r.=5/3
or (25/3)/5=25/(3*5)=5/3
or 125/9÷25/3=125/9×3/25=(125×3)/(9×25)=5/3
In a research article, you find that r is reported to be 4.8. How would you interpret this finding?
a. The relationship is reported incorrectly
b. The relationship is strong
c. The relationship is moderate
d. The relationship is weak
Based on the information given, the correct option is A. The relationship is reported incorrectly.
It should be noted that in a research, the normal value or r can range between -1 to 1. This shows the linear relationship between the variables.
In this case, since r is reported to be 4.8, the relationship is reported incorrectly.
Learn more about regression on:
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You would interpret this finding of r to be reported as 4.8 as (a) the relationship is reported incorrectly.
The variable r in regression represents correlation.
And in regression, correlation can only take values between -1 and 1 (inclusive)
Given that:
r = 4.8
4.8 is outside the range -1 to 1.
This means that, the value is either calculated incorrectly or reported incorrectly.
Hence, the true statement is (a)
Read more about regression and correlation at:
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what is a proportional relationship?
Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other.
Question one answer: S²=3V/H
What is the opposite operation of squaring? Using this opposite operation, rewrite the equation from question 1 so s is by itself on one side of the equation.
Step by step answer
Answer:
[tex]\ S=\sqrt{\dfrac{3V}{H}}}[/tex]
Step-by-step explanation:
The opposite operation of squaring is taking the square root.
[tex]\ S=\sqrt{\dfrac{3V}{H}}}[/tex]
We know that the denominator of a fractional power is the index of the corresponding root:
[tex]\displaystyle x^\frac{1}{n}=\sqrt[n]{x}[/tex]
For n=2, we don't usually write the index in the root symbol:
[tex]x^{\frac{1}{2}}=\sqrt{x}[/tex]
In the case of this problem, ...
[tex](S^2)^{\frac{1}{2}}=\left(\dfrac{3V}{H}\right)^{\frac{1}{2}}\\\\S=\sqrt{\dfrac{3V}{H}}[/tex]
A boxplot of weights (in grams) for 578 chicks is given. The right hand axis marks the five number summary. Use this plot to decide if the following statement is true or false. A chick weighing 300 grams is an outlier. [Note: use the Upper Fence criterion].a) trueb) false
Answer:
the anwser is in the text
Step-by-step explanation:
In choosing what music to play at a charity fund raising event, Cory needs to have an equal number of piano sonatas from J. S. Bach, Haydn, and Wagner. If he is setting up a schedule of the 9 piano sonatas to be played, and he has 5 J. S. Bach, 52 Haydn, and 5 Wagner piano sonatas from which to choose, how many different schedules are possible
Answer:
2,210,000 different schedules
Step-by-step explanation:
Cory needs to have an equal number of piano sonatas from J. S. Bach, Haydn, and Wagner.
Since he is setting up a schedule of 9 piano sonatas to be played, he needs:
3 out of 5 J. S. Bach piano sonatas3 out of 52 Haydn piano sonatas3 out of 5 Wagner piano sonatasWe then calculate how many different schedules are possible using combination.
Number of possible Schedules
[tex]=$ ^5C_3$ \times ^{52}C_3$ \times ^5C_3\\=10 \times 22100 \times 10\\$=2210000 ways[/tex]
There are 2,210,000 different possible schedules.
India is the second most populous country in the world, with a population in 2008 of about 1.14 billion people. The population is growing by about 1.34% each year. If the population continues following this trend, during what year will the population reach 2 billion?
Answer:
India's population will reach 2 billion during the year of 2050.
Step-by-step explanation:
India's population in t years after 2008 is modeled by the following equation:
[tex]P(t) = P(0)(1+r)^{t}[/tex]
In which P(0) is the population in 2008 and r is the growth rate, as a decimal.
Population in 2008 of about 1.14 billion people. The population is growing by about 1.34% each year.
This means that [tex]P(0) = 1.14, r = 0.0134[/tex]. So
[tex]P(t) = P(0)(1+r)^{t}[/tex]
[tex]P(t) = 1.14(1+0.0134)^{t}[/tex]
[tex]P(t) = 1.14(1.0134)^{t}[/tex]
If the population continues following this trend, during what year will the population reach 2 billion?
t years after 2008.
t is found when P(t) = 2. So
[tex]P(t) = 1.14(1.0134)^{t}[/tex]
[tex]2 = 1.14(1.0134)^{t}[/tex]
[tex](1.0134)^{t} = \frac{2}{1.14}[/tex]
[tex]\log{(1.0134)^{t}} = \log{\frac{2}{1.14}}[/tex]
[tex]t\log{1.0134} = \log{\frac{2}{1.14}}[/tex]
[tex]t = \frac{\log{\frac{2}{1.14}}}{\log{1.0134}}[/tex]
[tex]t = 42.23[/tex]
2008 + 42 = 2050
India's population will reach 2 billion during the year of 2050.
Two dice are rolled. E is the event that the sum is even, F is the event of rolling at least one six, and G is the event that the sum is eight. List the outcomes for the following events:
a. E ∩ F {(2, 2), (4, 4), (6, 6)} {(6, 2), (6, 4), (6, 6), (2, 6), (4, 6)} {(2, 6), (4, 6), (6, 6)} ∅
b. Ec ∩ G {(6, 2), (6, 4), (6, 6), (2, 6), (4, 6)} {(2, 6), (4, 6), (6, 6)} {(2, 2), (4, 4), (6, 6)} ∅
Answer:
(a)[tex]E \cap F ={(2, 6),(4, 6),(6, 2),(6, 4),(6, 6)}[/tex]
(b) [tex]E^c \cap G =\{ \}[/tex]
Step-by-step explanation:
The sample space of two dice rolled is given below:
[tex]\{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)\\(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)\\(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)\\(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)\\(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),(5,6)\\(6, 1), (6, 2), (6, 3), (6, 4), (6,5), (6,6)\}[/tex]
For Event E (The sum is even), the outcomes are:
[tex](1, 1), (1, 3), (1, 5),(2, 2), (2, 4), (2, 6)\\(3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6)\\(5, 1),(5, 3), (5, 5), (6, 2), (6, 4), (6,6)[/tex]
For Event F (Rolling at least one six), the outcomes are:
[tex](1, 6), (2, 6), (3, 6), (4, 6),(5,6),(6, 1), (6, 2), (6, 3), (6, 4), (6,5), (6,6)[/tex]
For Event G (The sum is eight), the outcomes are:
[tex](2, 6), (3, 5),(4, 4), (5, 3),(6, 2)[/tex]
(a)[tex]E \cap F[/tex]
Therefore:
[tex]E \cap F ={(2, 6),(4, 6),(6, 2),(6, 4),(6, 6)}[/tex]
(b)[tex]E^c \cap G[/tex]
E is the event that the sum is even
Therefore: [tex]E^c$ is the event that the sum is odd.[/tex]
Since G is the event that the sum is eight( which is even), the intersection of the complement of E and G will be empty.
Therefore:
[tex]E^c \cap G =\{ \}[/tex]
Some people say Mauna Kea is a taller mountain than Mt. Everest, because its total under sea and above sea level height is
a0 meters.
Answer:
Mauna Kea has from the base that is to say the deepest 10 km of height that is to say 10,000 m which is equivalent to being higher than Mount Everest.jj
Step-by-step explanation:
The main reason why we could say that Mauna Kea is the highest mountain, even surpassing Mount Everest, is because of its altitude if we compare them we see the difference; for example Mauna Kea from its base that is to say from the deepest the ocean measures 10,000 m of altitude, on the other hand the Mount Everest has an altitude of 8848 meters (29,029 ft) above sea level, that is to say from its base it measures 8848 m altitude.
Alexander threw a dart at this board a total of 40 times. Predict the number of times the dart will land on the number 3.
PLZ HURRY
Answer:
5% chance so 2 times
Step-by-step explanation:
Got chu:)
Find the roots of the polynomial function f(x)=x^3+2x^2+x
Answer:
0 and -1
Step-by-step explanation:
Hello,
[tex]x^3+2x^2+x = x(x^2+2x+1)=x(x+1)^2[/tex]
so the roots are
0 and -1 (to get x+1=0 )
do not hesitate if you need further explanation
hope this helps
Answer:
It's actually X = 0, X = - 1, with multiplicity 2, Option D, on Edge2020
Step-by-step explanation:
I just did it on edge :)
James grows corn on 1/4 of his land. If he has 65 acres of land, how much of the
land is used for growing corn?
acres
Answer:
16.25 acres
Step-by-step explanation:
If corn is 1/4 of the land and we have 65 acres, we merely multiply 1/4 to 65.
You get 16.25 acres.
Answer:
16.25acres
Step-by-step explanation:
65/4=16.25 or 16 and 1/4
Harriet spins this 6 colour spinner and flips a coin.
What is the probability of getting grey and tails?
Answer: 9%
Step-by-step explanation:
In probabilities; when two or more conditions must be met, the "and" operator is used. The probability of grey and tails is 1/12
From the attached spinner, we have:
[tex]Grey = 1[/tex]
[tex]n =6[/tex] --- partitions
So, the probability of landing on grey is:
[tex]P(Grey) = \frac{Grey}{n}[/tex]
This gives:
[tex]P(Grey) = \frac{1}{6}[/tex]
In a coin, we have:
[tex]Tail = 1[/tex]
[tex]n =2[/tex] --- faces
So, the probability of tail is:
[tex]P(Tail) = \frac{Tail}{n}[/tex]
[tex]P(Tail) = \frac{1}{2}[/tex]
The probability of grey and tail is:
[tex]Pr = P(Grey) \times P(Tail)[/tex]
[tex]Pr = \frac{1}{6} \times \frac{1}{2}[/tex]
[tex]Pr = \frac{1}{12}[/tex]
Hence, the probability of getting grey and tail is 1/12
Read more on probability at:
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What would the amplitude be? How do I find it? Should I add 1.25+6.75 then divide by 2 ?
Answer:
2.75
Step-by-step explanation:
The amplitude is half of the difference between the highest and lowest y-coordinates.
amplitude = 0.5|6.75 - 1.25| = 0.5(5.5) = 2.75
what are the cubic units? Pleaseee
Answer: pls mark me brainiest
Step-by-step explanation:
I am very sure that the answer is V≈351.86
Answer:
112π
Step-by-step explanation:
Cubic units are units when talking in the 3rd dimension
The volume of a cylinder is πr² x h
find the area of the base first.
since we can write it in terms of π, don't worry about decimals.
The radius is 4.
plug it in, the base's area is 16π. Multiply that by the height, 7.
16(7)π
112π is the volume
Jo and Dee are fairies. Jo is 888 centimeters tall. Dee is 777 centimeters taller than Jo.
How tall is Dee?
Answer:
15 cm
Step-by-step explanation:
7 cm taller than 8 cm is 15 cm tall.
Dee is 15 cm tall.
Answer:
Dee's height is
[tex]1665cm[/tex]
Step-by-step explanation:
[tex]jo = 888cm \\ dee = 777cm \: \: \: taller \: \: \: than \: \: jo \\ [/tex]
So
[tex]dee = 888 + 777 \\ = 1665cm[/tex]
hope this helps
brainliest appreciated
good luck! have a nice day!
The city of Ventura would like to build a seawall to protect the city from the threat of tsunamis. Each additional inches of height further protects the city and the 100 residents are each willing to pay $10 per inch of seawall height, regardless of how many inches are provided. The cost of building a wall that is i inches high is given by c(i) = 6i^2. What is the Pareto Optimal height for the seawall?
Answer:
The Pareto Optimal height is [tex]i = 100 \ inch[/tex]
Step-by-step explanation:
The Pareto Optimal height is a height of the seawall at which an increase in wall height will exceed the amount the resident are willing to pay and a decrease will affect the protection of the city
The number of residents is [tex]n = 100[/tex]
The amount each are willing to pay is [tex]z=[/tex]$10 per inch
The cost of building a wall that is i inches high is given by [tex]c(i) = 6i^2.[/tex]
The total amount the residents are willing to pay is
[tex]n = 100 * 10[/tex] = $1000
The maximum cost is mathematically represented as
[tex]\frac{dc(i)}{di} = 10i[/tex]
which implies that
1000 = 10i
Hence the Pareto Optimal height is
=> [tex]i = \frac{1000}{10}[/tex]
[tex]i = 100 \ inch[/tex]
Each contestant in the Hunger Games must be trained to compete. Suppose that the time it takes to train a contestant has mean 5 days and standard deviation 4 days, independent of the time it takes other contestants to train. If the Hunger Games has 100 contestants to train, approx imate the probability that it will take less than 450 days to train all the contestants. Leave your answer in terms of the standard normal distribution phi(a).
Answer:
11.51% probability that it will take less than 450 days to train all the contestants.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For the sum of n variables, the mean is [tex]\mu*n[/tex] and the standard deviation is [tex]s = \sigma\sqrt{n}[/tex]
In this question:
[tex]n = 100, \mu = 100*5 = 500, s = 4\sqrt{100} = 40[/tex]
Approximate the probability that it will take less than 450 days to train all the contestants.
This is the pvalue of Z when X = 450.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{450 - 500}{40}[/tex]
[tex]Z = -1.2[/tex]
[tex]Z = -1.2[/tex] has a pvalue of 0.1151
11.51% probability that it will take less than 450 days to train all the contestants.
Which statements are true about reflections? Check all that apply.
An image created by a reflextion will always be congruent to its pre-image.
An image and its pre-image are always the same distance from the line of reflection.
If a point on the pre-image lies on the line of reflection, the image of that point is the same as the pre-image.
The line of reflection is perpendicular to the line segments connecting corresponding vertices.
The line segments connecting corresponding vertices are all congruent to each other.
The line segments connecting corresponding vertices are all parallel to each other.
A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 31 ft/s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (Round your answer to one decimal place.) ft/s (b) At what rate is his distance from third base increasing at the same moment? (Round your answer to one decimal place.)
Answer:
Step-by-step explanation:
Given that :
the side of the square = 90ft
The speed of the runner = 31 ft/sec
By the time the runner is halfway to the first base; the distance covered by the runner in time(t) is (31 t) ft and the distance half the base = 90/2 = 45 ft
Thus; 31 t = 45
t = 45/31
From the second base ; the distance is given as:
P² = (90)² + (90 - 31t )²
P = [tex]\sqrt{(90)^2 + (90 - 31t )^2}[/tex]
By differentiation with time;
[tex]\dfrac{dP}{dt} =\dfrac{1}{ 2 \sqrt{90^2 +(90-31t)^2} } *(0+ 2 (90-31t)(0-31))[/tex]
[tex]\dfrac{dP}{dt} =\dfrac{1}{ 2 \sqrt{90^2 +(90-31t)^2} } * 2 (-31)(90-31t)[/tex]
At t = 45/31
[tex]\dfrac{dP}{dt} =\dfrac{1}{ 2 \sqrt{90^2 +45^2} } * 2 (-31)(45)[/tex]
[tex]\dfrac{dP}{dt} =\dfrac{-35*45}{100.623}[/tex]
= - 13.86 ft/sec
Hence, we can conclude that as soon as the runner is halfway to the first base, the distance to the second base is therefore decreasing by 13.86 ft/sec
b) The distance from third base can be expressed by the relation:
q² = (31t)² + (90)²
[tex]q = \sqrt{(31t)^2+(90)^2}[/tex]
By differentiation with respect to time:
[tex]\dfrac{dq}{dt} = \dfrac{1}{2\sqrt{90^2 + (31)t^2} } *(0+31^2 + 2t)[/tex]
At t = 45/31
[tex]\dfrac{dq}{dt} = \dfrac{1}{2\sqrt{90^2 + 45^2} } *(0+31^2 + \frac{45}{31})[/tex]
[tex]= \dfrac{31*45}{100.623}[/tex]
[tex]= 13.86 \ ft/sec[/tex]
Thus, the rate at which the runner's distance is from the third base is increasing at the same moment of 13.86 ft/sec. So therefore; he is moving away from the third base at the same speed to the first base)
a) The distance from second base is decreasing when the batter is halfway to first base at a rate of 13.9 feet per second.
b) The distance from third base is increasing when the batter is halfway to first base at a rate of 13.9 feet per second.
a) As the batter runs towards the first base, both the distance from second base and the length of the line segment PQ decrease in time. The distance from the second base is determined by Pythagorean theorem:
[tex]QS^{2} = QP^{2}+PS^{2}[/tex] (1)
By differential calculus we derive an expression for the rate of change of the distance from second base ([tex]\dot QS[/tex]), in feet per second:
[tex]2\cdot QS \cdot \dot{QS} = 2\cdot QP\cdot \dot{QP} + 2\cdot PS\cdot \dot {PS}[/tex]
[tex]\dot{QS} = \frac{QP\cdot \dot QP + PS\cdot \dot{PS}}{QS}[/tex]
[tex]\dot {QS} = \frac{QP\cdot \dot {QP}+PS\cdot \dot {PS}}{\sqrt{QP^{2}+PS^{2}}}[/tex] (2)
If we know that [tex]QP = 0.5L[/tex], [tex]PS = L[/tex], [tex]L = 90\,ft[/tex], [tex]\dot {QP} = -31\,\frac{ft}{s}[/tex] and [tex]\dot {PS} = 0\,\frac{ft}{s}[/tex], then the rate of change of the distance from second base is:
[tex]\dot {QS} = \frac{(45\,ft)\cdot \left(-31\,\frac{ft}{s} \right)}{\sqrt{(45\,ft)^{2}+(90\,ft)^{2}}}[/tex]
[tex]\dot {QS} \approx -13.864\,\frac{ft}{s}[/tex]
The distance from second base is decreasing when the batter is halfway to first base at a rate of 13.9 feet per second.
b) As the batter runs towards the first base, both the distance from third base increases and the distance from home increase in time. The distance from the third base is determined by Pythagorean theorem:
[tex]QT^{2} = HT^{2}+QH^{2}[/tex] (3)
By differential calculus we derive an expression for the rate of change of the distance from third base ([tex]\dot QT[/tex]), in feet per second:
[tex]2\cdot QT\cdot \dot{QT} = 2\cdot HT\cdot \dot {HT} + 2\cdot QH\cdot \dot {QH}[/tex]
[tex]\dot {QT} = \frac{HT\cdot \dot {HT}+QH\cdot \dot {QH}}{QT}[/tex]
[tex]\dot {QT} = \frac{HT\cdot \dot {HT}+QH\cdot \dot {QH}}{\sqrt{HT^{2}+QH^{2}}}[/tex]
If we know that [tex]HT = 90\,ft[/tex], [tex]QH = 45\,ft[/tex], [tex]L = 90\,ft[/tex], [tex]\dot{HT} = 0\,\frac{ft}{s}[/tex] and [tex]\dot {QH} = 31\,\frac{ft}{s}[/tex], then the rate of change of the distance from third base is:
[tex]\dot{QT} = \frac{(45\,ft)\cdot \left(31\,\frac{ft}{s} \right)}{\sqrt{(90\,ft)^{2}+(45\,ft)^{2}}}[/tex]
[tex]\dot{QT} \approx 13.864\,\frac{ft}{s}[/tex]
The distance from third base is increasing when the batter is halfway to first base at a rate of 13.9 feet per second.
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