Final answer:
Kim's original speed was 70 km/h. This was determined by equating the distances driven by both Kim and Maria in terms of Kim's original speed, and the fact they traveled for the same amount of time.
Explanation:
Let's denote Kim's original speed as [tex]\(V_{o}\)[/tex] in km/h. Kim drove for 3 hours at this speed and then slowed down by 15 km/h, driving at [tex]\(V_{o} - 15\)[/tex] km/h for the next 3 hours. Maria, on the other hand, averaged a speed of [tex]\(V_{o} + 5\)[/tex] km/h for the entire 6-hour trip (from 9:00 am to 2:00 pm).
To find the distance, which is the same for both Maria and Kim, we can set up the following equations based on the fact that distance is the product of speed and time: Kim's distance traveled is [tex]3 \(V_{o}\) + 3\((V_{o} - 15)\)[/tex] and Maria's distance traveled is [tex]5\((V_{o} + 5)\)[/tex]. These two expressions should be equal, as they traveled the same route:
[tex]3 \(V_{o}\) + 3\((V_{o} - 15)\) = 5\((V_{o} + 5)\)[/tex]
Simplifying the equation:
[tex]3 \(V_{o}\) + 3 \(V_{o}\) - 45 = 5 \(V_{o}\) + 25[/tex]
[tex]6 \(V_{o}\) - 45 = 5 \(V_{o}\) + 25[/tex]
[tex]Subtracting 6 \(V_{o}\) from both sides:[/tex]
[tex]\(V_{o}\) = 25+45[/tex]
Adding 45 to both sides:
[tex]\(V_{o}\) = 70[/tex]
Kim’s original speed, therefore, is 70 km/h.
The New Orleans Saints scored 7 fewer points than twice the points scored by the Pittsburgh Steelers. The two teams together scored a total of 32 points. Write and solve an equation to show how many points each team scored.
Answer:
(m) + 2 (m) - 7 = 32 is the needed expression.
Points scored by Pittsburgh Steelers = 13
Points scored by New Orleans Saints = 19
Step-by-step explanation:
Let us assume the points scored by Pittsburgh Steelers = m
So, the Points scored by New Orleans Saints
= 2 (Points scored by Pittsburgh Steelers ) - 7
= 2 (m) - 7
Also,the total points scored by both teams = 32
So the points scored by( New Orleans Saints + Pittsburgh Steelers) = 32
⇒ (m) + 2 (m) - 7 = 32
or, 3 m = 32 + 7 = 39
⇒ m = 39/ 3 = 13, or m = 13
So, the points cored by Pittsburgh Steelers = m = 13
and the points scored by New Orleans Saints = 2m - 17
= 2(13) - 7 = 19
The percent of working students increased by 14.1% to 31.7%, what was the percent prior to the increase? (Express your answer rounded correctly to the nearest tenth of a percent!)
Answer:
124.8 %
Step-by-step explanation:
Given,
Initial percentage of working students = 14.1%,
Final percentage of working students = 31.7%,
Thus, the percentage increase = [tex]\frac{\text{Final percentage-initial percentage}}{\text{Initial percentage}}\times 100[/tex]
[tex]=\frac{31.7 - 14.1}{14.1}\times 100[/tex]
[tex]=\frac{17.6}{14.1}\times 100[/tex]
[tex]=\frac{1760}{14.1}[/tex]
≈ 124.8 %
Final answer:
To find the original percentage of working students before an increase of 14.1%, subtract 14.1% from the new percentage of 31.7%, which gives us 17.6% as the original percentage.
Explanation:
The student is asking to find the original percentage of working students before a 14.1% increase resulted in a new percentage of 31.7%. To determine the original percentage, we need to subtract the increase from the final percentage. Here's the calculation:
Final percentage = Original percentage + Increase
31.7% = Original percentage + 14.1%
To find the original percentage, we subtract 14.1% from 31.7%:
Original percentage = 31.7% - 14.1% = 17.6%
Rounded to the nearest tenth of a percent, the original percentage of working students was 17.6%.
A plane flies at 190 mph for the first and last half hour of a flight. It flies at a higher constant speed for the rest of the flight. The route is 2090 miles. The plane is 990 miles from the destination 2 hours before the end of the flight.How many hours is the entire flight?
Answer:
4.18 hours
Step-by-step explanation:
Since the plane needs 2 hours to cover the las 990 miles at the end of the flight, and in the last half hour it flies with a speed of 190mph, meaning it covers a distance of 190 / 2 = 95 miles in this last half hours.
That means for the other 1.5 hours, its cruising speed is
[tex] v_c = \frac{990 - 95}{1.5} = 596.7 mph[/tex]
Beside the first and last half hour of the trip (which covers 190 miles), the cruising distance must be
d = 2090 - 190 = 1900 miles
The cruising time is therefore
[tex]t_c = \frac{d}{v_c} = \frac{1900}{596.7} = 3.18 hours[/tex]
Hence the total time for the entire flight is
3.18 + 1 = 4.18 hours
In 4.18 hours is the entire flight.
Given
A plane flies at 190 mph for the first and last half hours of a flight.
It flies at a higher constant speed for the rest of the flight.
The route is 2090 miles.
The plane is 990 miles from the destination 2 hours before the end of the flight.
Speed;The plane needs 2 hours to cover the last 990 miles at the end of the flight, and in the last half hour it flies with a speed of 190mph,
Then,
The distance covered by plane in last half hour is;
[tex]\rm Distance=\dfrac{190}{2}\\\\Distance=95 \ miles[/tex]
The velocity is;
[tex]\rm Velocity = \dfrac{Distance }{Time}\\\\Velocity=\dfrac{900-95}{1.5}\\\\Velocity=\dfrac{805}{1.5}\\\\Velocity=596.7[/tex]
The first and last half hour of the trip (which covers 190 miles), the cruising distance must be;
Distance = 2090 - 190 = 1900 miles
Therefore,
The cruising time is;
[tex]\rm Time =\dfrac{Distance}{Velocity}\\\\Time=\dfrac{1900}{596.7}\\\\Time=3.18 \ hours[/tex]
The total time for the entire flight is
= 3.18 + 1 = 4.18 hours
Hence, in 4.18 hours is the entire flight.
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Consider the sequence:
3 , 8 , 13 , 18 , 23....
The recursive formula for this sequence is:
[tex]a_{n} =a_{n-1} +5.[/tex]
In complete sentences, explain what [tex]a_{n},a_{n-1},[/tex] and the 5 represnt in the formula. Find [tex]a_{g}[/tex]. What do you need to know in order to find [tex]a_{g}[/tex] ?
The [tex]a_n[/tex] reprsents the nth term where n is some positive whole number {1,2,3,...}
The [tex]a_{n-1}[/tex] represents the term just before the nth term. For example, if n = 22 then [tex]a_{n} = a_{22}[/tex] and [tex]a_{n-1} = a_{21}[/tex]
The +5 at the end means we add 5 to the previous term just before the nth term to get the nth term. In other words, the rule is "add 5 to each term to get the next term".
To get the 9th term [tex]a_{9}[/tex], we need to find the terms before this one because the recursive sequence builds up. The 9th term depends on the 8th term, which depends on the 7th term, and so on. The countdown stops until you reach the first term.
-------
[tex]a_{1} = 3[/tex] (given)
[tex]a_{2} = 8[/tex] (given)
[tex]a_{3} = 13[/tex] (given)
[tex]a_{4} = 18[/tex] (given)
[tex]a_{5} = 23[/tex] (given)
[tex]a_{6} = a_{5}+5 = 23+5 = 28[/tex] (add 5 to the prior term)
[tex]a_{7} = a_{6}+5 = 28+5 = 33[/tex]
[tex]a_{8} = a_{7}+5 = 33+5 = 38[/tex]
[tex]a_{9} = a_{8}+5 = 38+5 = 43[/tex]
So the 9th term is [tex]a_{9} = 43[/tex]
It will travel 21 miles upstream in three hours going down stream we can show 39 miles in the same amount of time find the speed of the current in the speed of the boat in Still water
Answer: The speed of the current is 3 miles per hour.
The speed of the boat in still water is 10 miles per hour
Step-by-step explanation:
Let the speed of the boat be x miles per hour
Let the speed of the current be y miles per hour
Speed = distance / time
If the boat moves 39 miles downstream in 3 hours, then, the speed is
39/3= 13 miles per hour
Let us assume that it moved it moved in the same direction with the current hence, it moved in still water
The total speed would be x + y. Therefore
x + y = 13 - - - - - - -1
If the boat moves 21 miles upstream in 3 hours, the the speed is
21/ 3 = 7 miles per hour
It is assumed that it moved in the opposite direction to the current.
The total speed would be x - y. Therefore
x - y = 7 - - - - - - -2
Adding equation 1 and equation 2, it becomes
2x = 20
x = 20/2 = 10 miles per hour
y = 13 - x
y = 13 - 10 = 3 miles per hour
A professional baseball player signs a contract with a beginning salary of $3,000,000 for the first year and an annual increase of 4% per year beginning in the second year. That is, beginning in year 2, the athlete's salary will be 1.04 times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.
To find the athlete's salary for year 7 of the contract, apply a 4% annual increase to the previous year's salary starting from year 2. The athlete's salary for year 7 is approximately $3,836,192.
Explanation:To find the athlete's salary for year 7 of the contract, we need to calculate the annual increase for each year from year 2 to year 7 and apply it to the starting salary of $3,000,000 for year 1.
In year 2, the salary is 1.04 times the previous year's salary, so the salary for year 2 is $3,000,000 * 1.04 = $3,120,000.
In year 3, the salary is 1.04 times the previous year's salary, so the salary for year 3 is $3,120,000 * 1.04 = $3,244,800.
Following the same pattern, we can calculate the salaries for years 4, 5, 6, and 7:
In year 4: $3,244,800 * 1.04 = $3,379,392
In year 5: $3,379,392 * 1.04 = $3,523,027.68
In year 6: $3,523,027.68 * 1.04 = $3,675,348.11
In year 7: $3,675,348.11 * 1.04 = $3,836,191.72
Therefore, the athlete's salary for year 7 of the contract is approximately $3,836,192.
The athlete's salary for year 7 of the contract is $3,822,736.
To calculate the athlete's salary for year 7, we use the formula for compound interest, which is also applicable to salaries that increase annually at a fixed percentage. The formula is:
[tex]\[ A = P(1 + r)^n \][/tex]
Given that the initial salary (principal amount P is $3,000,000 and the annual increase rate r is 4% (or 0.04 as a decimal), we can calculate the salary for year 7 as follows:
[tex]\[ A = 3,000,000(1 + 0.04)^{7-1} \] \[ A = 3,000,000(1.04)^6 \] \\[ A = 3,000,000 \times 1.26530612 ](after calculating ( 1.04^6 \)) \\\\[A = 3,822,736 \][/tex]
(after rounding to the nearest dollar)
Therefore, the athlete's salary for year 7 of the contract, rounded to the nearest dollar, is $3,822,736.
From base camp, a hiker walks 3.5 miles west and 1.5 miles north. Another hiker walks 2 miles east and 0.5 miles south. To the nearest tenth of a mile how far apart are the hikers
Answer:
The hikers are 5.9 miles apart.
Step-by-step explanation:
Let O represents the base camp,
Suppose after walking 3.5 miles west, first hiker's position is A, then after going 1.5 miles north from A his final position is B,
Similarly, after walking 2 miles east, second hiker's position is C then going towards 0.5 miles south his final position is D.
By making the diagram of this situation,
Let D' is the point in the line AB,
Such that, AD' = CD
In triangle BD'D,
BD' = AB + AD' = 1.5 + 0.5 = 2 miles,
DD' = AC = AO + OC = 3.5 + 2 = 5.5 miles,
By Pythagoras theorem,
[tex]BD^2 = BD'^2 + DD'^2[/tex]
[tex]BD = \sqrt{2^2 + 5.5^2}=\sqrt{4+30.25}=\sqrt{34.25}\approx 5.9[/tex]
Hence, the hikers are 5.9 miles apart.
"Express as a mixed numeral: 39/8" is a problem found in a fifth-grade arithmetic textbook. This problem is BEST solved through: Please choose the correct answer from the following choices, and then select the submit answer button.
a.trial and error.
b.insight.
c.an algorithm.
d.a heuristic.
Answer:
c. an algorithm
Step-by-step explanation:
Most math problems are best solved using an algorithm. Certainly, converting numbers from one form to another is easily done that way.
_____
The integer part is the whole-number quotient of the division. The fractional part is the remainder divided by the denominator:
39/8 = 32/8 + 7/8 = 4 7/8
The average cost per month of a 2-bedroom apartment in Grayson was $625 last year.This year,the average cost is $650.What is the percent of increase from late year?
The amount increased 4% from last year.
Step-by-step explanation:
Given,
Average cost last year = $625
Average cost this year = $650
Increase amount = Average cost last year - Average cost this year
Increase amount = 650 - 625 = $25
Percent increase = [tex]\frac{Increase\ amount}{Average\ cost\ last\ year}*100[/tex]
[tex]Percent\ increase=\frac{25}{625}*100\\Percent\ increase=\frac{2500}{625}\\Percent\ increase=4\%[/tex]
The amount increased 4% from last year.
Keywords: percentages, subtraction
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One month melissa rented 12 movies and 2 video games for a total of $29.The next month she rented 3 movies and 5 video games for a total of 32$. Find the cost for each movie and each video game.
Answer:the cost of renting one movie is $1.5
the cost of renting one video game is $5.2
Step-by-step explanation:
Let x represent the cost of renting one movie.
Let y represent the cost of renting one video game. One month melissa rented 12 movies and 2 video games for a total of $29. This means that
12x + 2y = 29 - - - - - - -1
The next month she rented 3 movies and 5 video games for a total of 32$. This means that
3x + 5y = 32 - - - - - - - - 2
Multiplying equation 1 by 5 and equation 2 by 2, it becomes
60x + 10y = 145
6x + 10y = 64
Subtracting,
54x = 81
x = 81/54 = 1.5
Substituting x = 1.5 into equation 2, it becomes
3x + 5y = 32
3×2 + 5y = 32
5y = 32 - 6 = 26
y = 26/5 = 5.2
--------------------------------------------------------------
Answer: m<YSH = 35°
m<YIH = 20°
m<SAI = 71°
m<PAY = 8°
PY = i don' t know
and P.S these answers may all be wrong sorry but hope it helps. :)
Answer:
Step-by-step explanation:
What is the value of sin C ?
A. 8/17
B. 15/8
C. 15/17
D. 8/15
Answer:
The answer to your question is letter A. [tex]\frac{8}{17}[/tex]
Step-by-step explanation:
Sin C = [tex]\frac{opposite side }{hypotenuse}[/tex]
Opposite side = 8
hypotenuse = 17
Substitution and result
sin C= [tex]\frac{8}{17}[/tex]
Find the value of the given trigonometric function by finding the reffende angle x and the attaching proper sign Tan 287° What is the representation of tan 287° using reference angle x
Answer:
-tan(73°)
Step-by-step explanation:
The reference angle is the smallest angle between the terminal ray and the x-axis. Here, the terminal ray is in the 4th quadrant, so the reference angle will be ...
ref∠ = 360° -287° = 73°
In the 4th quadrant, the tangent is negative, so we have ...
tan(287°) = -tan(73°)
The volume V, in liters, of air in the lungs during a five-second respiratory cycle is approximated by the model V + 0.1729t + 0.1522t² - 0.0374t³, where t is the time in seconds. Approximate the average volume of air in the lungs during one cycle.
Answer:
The average volume of air in the lungs during one cycle is 0.53176 liters.
Step-by-step explanation:
Given : The volume V, in liters, of air in the lungs during a five-second respiratory cycle is approximated by the model [tex]V=0.1729t+0.1522t^2- 0.0374t^3[/tex], where t is the time in seconds.
To find : The average volume of air in the lungs during one cycle ?
Solution :
The volume V of air in the lungs during a five-second respiratory cycle is [tex]V=0.1729t+0.1522t^2- 0.0374t^3[/tex]
Then the average volume of air in the lungs during one cycle [0,5] is
[tex]V_a=\frac{1}{b-a}\int\limits^a_b {V(t)} \, dt[/tex]
[tex]V_a=\frac{1}{5-0}\int\limits^0_5 {0.1729t+0.1522t^2- 0.0374t^3} \, dt[/tex]
[tex]V_a=\frac{1}{5}[\frac{0.1729t^2}{2}+\frac{0.1522t^3}{3}- \frac{0.0374t^4}{4}}]^5_0[/tex]
[tex]V_a=\frac{1}{5}[0.08645t^2+0.05073t^3- 0.00935t^4]^5_0[/tex]
[tex]V_a=\frac{1}{5}[0.08645(5)^2+0.05073(5)^3- 0.00935(5)^4-0.08645(0)^2+0.05073(0)^3- 0.00935(0)^4][/tex]
[tex]V_a=\frac{1}{5}[2.1613+6.3413-5.8438][/tex]
[tex]V_a=\frac{1}{5}[2.6588][/tex]
[tex]V_a=0.53176\ l[/tex]
The average volume of air in the lungs during one cycle is 0.53176 liters.
Final answer:
To find the average volume of air in the lungs during one cycle, calculate the definite integral of V(t) over the interval t = 0 to t = 5 seconds.
Explanation:
The average volume of air in the lungs during one cycle can be approximated by finding the average value of the function V(t) over the interval t = 0 to t = 5 seconds. To do this, we need to find the definite integral of V(t) over the given interval:
∫[0,5] V(t) dt
After calculating the integral, divide the result by the length of the interval (5 seconds) to find the average volume of air in the lungs during one cycle.
A company declares a 5% stock dividend. The debit to retained earnings is an amount equal to
Answer:
A company declares a 5% stock dividend.
The debit to retained earnings is an amount equal to - the market value of the shares, that are to be issued.
We can say that a retained earnings balance is increased when we are using a credit and this is decreased when we make a debit.
A retained earnings is the total amount of money left, after all the expenses and dividends are paid by the company.
A 5% stock dividend represents a distribution of extra shares, 5% of existing ones, to shareholders. The debit to retained earnings is equal to 5% of the total market value of the previously existing shares.
Explanation:When a company declares a 5% stock dividend, this means that the company is distributing additional shares of its stock to the existing shareholders. The quantity of these additional shares is equal to 5% of the shares that each shareholder currently owns. The debit to retained earnings is an amount equal to the total market value of these additional shares, which is determined by multiplying the declared dividend rate (5%) by the total market value of circulating shares prior to the issuance of dividend.
For instance, if a company has 1000 shares with a market value of $10 each, it would issue additional 50 shares (5% of 1000 shares). Therefore, the debit to retained earnings would be $500 (50 shares * $10/share).
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find the domain od the given function.
f(x)= [tex]\frac{x}{x-3}[/tex]
a.) all real numbers
b.) (-∞,3)∪(3,∞)
c.) (0,∞)
d.) (-∞,-3)∪(-3,∞)
Answer:
B
Step-by-step explanation:
There is a vertical asymptote at x=3. It is not defined and therefore not included in the domain.
All other numbers are included in the domain.
[tex]f(x)=\frac{x+9}{4} \\g(x)=4x-9[/tex]
Show your work to prove that the inverse of f(x) is g(x)
Answer:
inverse of f(x) is g(x)
Step-by-step explanation:
f(x) = (x+9)/4
g(x) = 4x-9
f^-1(x) = g(x)
Let y = (x+9)/4
4y = x + 9 (make x the subject)
x = 4y - 9
f^-1(x) = 4x - 9
= g(x) (shown)
Answer:
Below.
Step-by-step explanation:
If f(x) is the inverse then f(g(x)) = x.
f(g(x)) = ((4x - 9) + 9) / 4
= 4x/4
= x.
So it is true.
It can also be shown that g(f(x)) = x.
yuh girl stupid help me out
when at rest an elephants heart rate is about 30 beats per min and a humans heart rate about 70 bpm. a humans heart beats about how many more times per day than an elephants heart
a. 960
b. 2400
c. 5760
d. 28800
e. 57600
Answer:
A. 960
Step-by-step explanation:
Multiply each heart rate by 24 (hours in a day)
Subtract the human's daily heart rate from the elephant's.
30 x 24 = 720
70 x 24 = 1680
1680 - 720 = 960
Answer:
It’s A
Step-by-step explanation:
Suppose a seventh graders birthday is today, and she is 12 years old. How old was she 3 1/2 years ago? Write an equation, and use a number line to model your answer
Answer:
he age of girl 3 [tex]\frac{1}{2}[/tex] years ago is 8.5 years .
Step-by-step explanation:
Given as :
The present age of seventh grader girl = 12 years
Let The age of her 3 [tex]\frac{1}{2}[/tex] years ago = x years
So The age of girl [tex]\frac{7}{2}[/tex] years ago = x years
Now, According to question
The age of girl 3.5 years ago = 12 - 3.5
Or, The age of girl 3.5 years ago = 8.5 years
Hence The age of girl 3 [tex]\frac{1}{2}[/tex] years ago is 8.5 years . Answer
Please help!!
Evaluate:
1. 5x(x-2)-2x^2 for x=-2
2. 2x^2-5x+8 for x=3
Answer:
Step-by-step explanation:
1) 5x(x-2)-2x^2 for x =2
= 5(-2)(-2-2) -2(2)^2
= -10(-4) -2(4)
= 40 - 8
= 36
2) 2x^2 - 5 + 8 for x = 3
= 2(3)^2 - 5(3) + 8
= 2(9) - 15 + 8
= 18 - 15 + 8
= 3 + 8
= 11
Answer:
1. 32.
2. 11.
Step-by-step explanation:
1. 5x(x-2)-2x^2 for x=-2:
= 5(-2)( -2-2)-2(-2)^2
= -10*-4 - 2*4
= 40-8
= 32.
2. 2x^2-5x+8 for x=3:
= 2(3)^2 -5(3)+8
= 18 - 15 + 8
= 11.
The complete outside including the bottom of a wooden 4 inch cube is painted red. The painted cube is then cut into 1 inch cubes . How many of the 1 inch cubes do not have red on any face?
Answer:
8
Step-by-step explanation:
Only the innermost 8 1-inch cubes would have no red paint on any face.
The 4 inch cube is divided into 1 inch cubes, and the number of cubes that do not have red on any face is calculated. There are no 1 inch cubes that do not have red on any face.
Explanation:To find the number of 1 inch cubes that do not have red on any face, we need to consider the number of cubes that have red on at least one face. Since the complete outside of the 4 inch cube is painted red, each face of the cube has a 4x4 square painted red. Therefore, the area of each face is 16 square inches.
Dividing the 4 inch cube into 1 inch cubes, we have a total of 64 smaller cubes (4x4x4 = 64). Each smaller cube has 6 faces.
So, the total area of all the faces of the smaller cubes is 64 x 6 = 384 square inches. Since each face of the smaller cubes has an area of 1 square inch, the number of cubes that have red on at least one face is 384/1 = 384 cubes.
Therefore, the number of 1 inch cubes that do not have red on any face is 64 - 384 = -320. However, since negative numbers do not make sense in this context, we can conclude that there are no 1 inch cubes that do not have red on any face.
Awesome autos car detail charges $30 for a full service detail on SUVs. The equation 30x+45y=450 represents the amount that must be earned each hour to cover expenses, where x represents cars and y represents SUVs. Determine the x and y intercepts of this equation and state what each one means in terms of business
Answer:
x intercept = 15 # of cars to detail if no SUVs
y intercept = 10 # of SUVs to detail if no cars
Hope this helps!
The x-intercept and the y-intercept for the equation 30x + 45y = 450 are 15 and 10 respectively, The x-intercept denotes the number of cars if there is no SUV serviced and the y-intercept represents the number of SUVs if there is no car.
What is the equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Given:
Awesome autos car detail charges $30 for a full-service detail on SUVs,
The equation 30x + 45y = 450 represents the amount that must be earned each hour to cover expenses,
From this equation, we can determine that the autos car detail charges $45 for a full-service detail on car,
Find the x-intercept as shown below,
30x + 45 × 0 = 450 (For x-intercept y = 0)
30x = 450
x = 450 / 30
x = 15
Find the y-intercept as shown below
30 × 0 + 45y = 450 (For y-intercept x = 0)
45y = 450
y = 10
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The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 120I I2 + I + 1 where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?
Answer:
Light intensity = L =1
Step-by-step explanation:
The given function is:
[tex]P = \frac{120L}{L^{2}+L+1 }[/tex] (Equation.1)
Taking derivative of the whole equation:
[tex]\frac{dP}{dL}[/tex] =[tex]\frac{d}{dL}[/tex] [tex](\frac{120L}{L^{2}+L+1 } )[/tex]
[tex]\frac{dP}{dL}[/tex] =120 [tex]\frac{1 (L^{2}+L+1) - (\frac{d}{dL}L^{2} +\frac{d}{dL} L+\frac{d}{dL}1)L}{(L^{2}+L+1)^{2} }[/tex]
[tex]\frac{dP}{dL}[/tex] =[tex]\frac{120 (L^{2}-1)}{L^{2}+L+1 }[/tex]
[tex]\frac{dP}{dL}[/tex] =[tex]\frac{120 (L+1)(L-1)}{L^{2}+L+1 }[/tex]
For first derivative test, there is a maxima. for that , we need to take:
L+1 =0 and L-1 =0
we get:
L= -1
or L=1
Since light intensity cannot be negative so, L=1
put this in equation 1:
[tex]P(1) = \frac{120}{1^{2}+1+1 }[/tex]
P = [tex]\frac{120}{3}[/tex]
P= 40 which is the maximum value of P (At L=1)
To find the light intensity that maximizes photosynthesis, you must differentiate the function P = 120I/(I^2 + I + 1) with respect to I, equate it to zero and solve for I. Then, use these values to examine the second derivative and verify whether these are local maxima or minima.
Explanation:To find the light intensity (I) that maximizes photosynthesis rate (P), we should analyze the function P = 120I/(I2 + I + 1) using calculus concepts. This type of problem is looking for a maximum value, so we need to find where the derivative of the function equals zero.
First, differentiate the function P regarding I, using quotient rule: P' = (120(I² + I + 1) - 120I (2I + 1))/ (I² + I + 1)² = 0.
Then, solve this equation for I to find the values for which the rate of change of P with respect to I is zero. These are the points where P is at a maximum or a minimum. Finally, check the second derivative of the function P on these values to determine whether these correspond to a local maximum, a local minimum, or a saddle point.
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From 27 pieces of luggage, an airline luggage handler damages a random sample of four. The probability that exactly one of the damaged pieces of luggage is insured is twice the probability that none of the damaged pieces are insured. Calculate the probability that exactly two of the four damaged pieces are insured.
Answer:
0.273
Step-by-step explanation:
Let the number of insured pieces of luggage be i and u be the number of uninsured pieces of luggage, therefore,
i + u = 27
Now,
probability that exactly one of the damaged pieces of luggage is insured = (iC1)(uC3)/(27C4)
probability that none of the damaged pieces are insured = (uC4)/(27C4)
and,
(iC1)(uC3)/(27C4) = 2 (uC4)/(27C4)
=> u − 2i = 3
By solving, i + u = 27 and u − 2i = 3
i = 8 and u = 19
and,
(8C2)(19C2)/(27C4) = 0.273
Answer:
Step-by-step explanation:
Solve x2 - 4 = 5 by graphing the related function,
There are two solutions: #1,
olve 80%
There are two solutions: 3 and -3.
There is one solution: 1
quations
There are no real number solutions.
quare
its B. on ed :)
Hope this helps
Simplify.
(3x2−2x+2)−(x2+5x−5)
a.4x^2+3x−3
b.2x^2+3x−3
c.2x^2−7x+7
d.2x^2−3x−3
Answer:
C. 2x^2-7x+7
Step-by-step explanation:
(3x^2-2x+2)-(x^2+5x-5) Given
From the given you will subtract like terms from both parenthesis.
3x^2-x^2=2x^2
-2x-5x=-7x
2-(-5)=7
To simplify the given expression, distribute the negative sign through the second parentheses, combine like-terms, which results in 2x^2 - 7x + 7, corresponding to option (c).
Explanation:To simplify the expression (3x2−2x+2)−(x2+5x−5), you'd start by distributing the negative sign through the second parentheses.
Doing so gives us 3x^2 - 2x + 2 - x^2 - 5x + 5.
Upon combining like-terms, the expression becomes 2x^2 - 7x + 7.
So, the simplified form of the original expression is 2x^2 - 7x + 7.
In the given options,
c) 2x²-7x+7 is right
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Suppose the number of leaves z on a tree varies directly with the height of the tree y in yards and directly with the cube of the hours x of direct sunlight that the tree receives. If a tree that is 7 yards tall and receives 6 hours of direct sunlight has 6048 leaves on it, then how many leaves will be on a tree that is 2 yards tall but receives 11 hours of sunlight?
Answer:
The number of leaves are 10648
Step-by-step explanation:
By the given information,
number of leaves be= z
height of tree= y
hours of sunlight= x
by the given conditions,
[tex]z = (k)(y)(x^{3})[/tex]
,where k is portionality constant.
now, z=6048, y=7 and x=6,
evaluating value of k using above conditions,
k=4
thus, using this value of k,
[tex]z=(4)(2)(11^{3})[/tex]
z= 10648.
The number of leaves on a tree of 2 yards tall receiving 11 hours of sunlight is predicted to be approximately 8712 based on the given relation.
Explanation:The number of leaves z on a tree varies directly with the height of the tree y in yards and directly with the cube of the hours x of direct sunlight that the tree receives. Let's represent this relationship as z = kyx³ , where k is a constant of proportionality.
From the information provided, we know that a tree that is 7 yards tall and receives 6 hours of direct sunlight has 6048 leaves. Therefore, we can substitute these values into the equation and solve for k. Thus, if we replace z = 6048, y = 7, and x = 6, we find that k is approximately equal to 3.
With k approximately equal to 3, we can then predict the number of leaves on a tree that is 2 yards tall and receives 11 hours of sunlight. If we replace k = 3, y = 2, and x = 11 into our equation, we find that the predicted number of leaves is approximately 8712.
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A cup of coffee with temperature 155degreesF is placed in a freezer with temperature 0degreesF. After 5 minutes, the temperature of the coffee is 103degreesF. Use Newton's Law of Cooling to find the coffee's temperature after 15 minutes.
The coffee's temperature after 15 minutes in the freezer, using Newton's Law of Cooling with a cooling constant [tex]\( k \approx 0.0816 \),[/tex] is approximately [tex]\( 45.57^\circ F \).[/tex]
Newton's Law of Cooling is given by the equation:
[tex]\[ T(t) = T_a + (T_0 - T_a) \cdot e^{-kt} \][/tex]
Where:
[tex]\( T(t) \)[/tex]= temperature of the object at time \( t \)
[tex]\( T_0 \)[/tex] = initial temperature of the object
[tex]\( T_a \)[/tex] = ambient temperature
[tex]\( k \)[/tex] = cooling constant
[tex]\( t \)[/tex] = time
Given:
[tex]\( T_0 = 155^\circ F \)[/tex] (initial temperature)
[tex]\( T_a = 0^\circ F \)[/tex] (ambient temperature)
[tex]\( T(5) = 103^\circ F \)[/tex](temperature after 5 minutes)
We need to find the coffee's temperature[tex](\( T(15) \))[/tex] after 15 minutes. First, we need to determine the cooling constant [tex]\( k \)[/tex].
Using the given information, let's rearrange Newton's Law of Cooling to solve for [tex]\( k \)[/tex]:
[tex]\[ T(t) = T_a + (T_0 - T_a) \cdot e^{-kt} \][/tex]
[tex]\[ T(t) - T_a = (T_0 - T_a) \cdot e^{-kt} \][/tex]
[tex]\[ \frac{T(t) - T_a}{T_0 - T_a} = e^{-kt} \][/tex]
[tex]\[ -kt = \ln\left(\frac{T(t) - T_a}{T_0 - T_a}\right) \][/tex]
[tex]\[ k = -\frac{1}{t} \cdot \ln\left(\frac{T(t) - T_a}{T_0 - T_a}\right) \][/tex]
Given [tex]\( T(5) = 103^\circ F \)[/tex]:
[tex]\[ k = -\frac{1}{5} \cdot \ln\left(\frac{103 - 0}{155 - 0}\right) \][/tex]
[tex]\[ k = -\frac{1}{5} \cdot \ln\left(\frac{103}{155}\right) \][/tex]
[tex]\[ k \approx -\frac{1}{5} \cdot \ln(0.6645) \][/tex]
[tex]\[ k \approx -\frac{1}{5} \cdot (-0.4081) \][/tex]
[tex]\[ k \approx 0.0816 \][/tex]
Now that we have the cooling constant [tex]\( k \)[/tex], we can find [tex]\( T(15) \)[/tex]:
[tex]\[ T(15) = T_a + (T_0 - T_a) \cdot e^{-kt} \][/tex]
[tex]\[ T(15) = 0 + (155 - 0) \cdot e^{-0.0816 \cdot 15} \][/tex]
[tex]\[ T(15) = 155 \cdot e^{-1.224} \][/tex]
[tex]\[ T(15) \approx 155 \cdot 0.294 \][/tex]
[tex]\[ T(15) \approx 45.57^\circ F \][/tex]
Therefore, the coffee's temperature after 15 minutes in the freezer is approximately [tex]\( 45.57^\circ F \)[/tex].
What is the equation of the line 2-3y = 18 in slope-intercept form?
A drawer holds 6 white socks, 8 black socks, and 4 blue socks. If a sock is pulled from the drawer randomly, what is the probability that it will not be a blue sock?
Answer:
P(No blue)=7/9=0.778
Step-by-step explanation:
The sample space of an experiment is "the set of all possible outcomes of that experiment".
The Complement Rule states "that the sum of the probabilities of an event and its complement must equal 1". So for example if A is an event and A' their complement we have this:
[tex]P(A)+P(A')=1[/tex]
[tex]P(A')=1-P(A)[/tex]
The probability of an event is "a number describing the chance that the event will happen".
For this problem we need to find first the sample space and is given by:
[tex]\Omega = [6white,8Black,4Blue][/tex] for a total of 18 socks.
[tex]n(\Omega)=8[/tex]
We can use the definition of probability of an event, and we find the probability that the randomly select sock would be blue we have:
[tex]P_{blue}=\frac{Favorable outcomes}{Total outcomes}=\frac{4}{18}=\frac{2}{9}[/tex]
And then using the complement rule if we want the probability that the randomly selected sock would be NOT blue we have:
[tex]P'_{blue}=1-P_{blue}=1-\frac{2}{9}=7/9=0.778[/tex]