The time at which the population reaches 4,000 is t ≈ 2.51 hours after 12 pm or approximately 2:30 pm. A) The growth constant can be found using the formula for exponential growth:
$-N = N_0 e^{kt}$
where N₀ is the initial population, N is the final population, t is the time elapsed, and k is the growth constant.
Using the given information, we can set up two equations:
500 = N₀e^(0k)
1500 = N₀e^(2k)
Dividing the second equation by the first, we get:
3 = e^(2k)
Taking the natural logarithm of both sides, we get:
ln(3) = 2k
Therefore, the growth constant k is (ln(3))/2, approximately 0.549.
The population as a function of time can now be expressed as:
N(t) = 500e^(0.549t)
B) To find the population at 5 pm, we need to substitute t = 5 into the equation we found in part A:
N(5) = 500e^(0.549*5) ≈ 4,206
Therefore, the population at 5 pm is approximately 4,206 bacteria.
C) To find the time the population reaches 4,000, we need to solve the equation N(t) = 4,000 for t:
4,000 = 500e^(0.549t)
Dividing both sides by 500, we get:
8 = e^(0.549t)
Taking the natural logarithm of both sides, we get:
ln(8) = 0.549t
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Melissa is tracking the progress of her plants growth. She measures the plant and finds that it is 5. 5 inches tall. The plant then is measured every day for the next few days and she finds it is growinf at a constant rate of 1. 4 inches per day. What is the y-intercept
The y-intercept is 5.5 inches, which is the initial height of the plant before it started growing.
We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept, to solve the problem.
Since the plant is growing at a constant rate of 1.4 inches per day, the slope of the line that represents its growth is 1.4.
Let y be the height of the plant after x days. We know that the plant was 5.5 inches tall on day 0, so we have the point (0, 5.5) on the line.
Using the point-slope form of a linear equation, we get:
y - 5.5 = 1.4x
Simplifying, we get:
y = 1.4x + 5.5
Comparing the equation to the slope-intercept form, we see that the y-intercept is 5.5.
Therefore, the y-intercept is 5.5 inches.
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PLEASE HELP ASAP!!!!!!!!
Joseph bought a pair of socks for $9, a pair of shoes for $40, and a pair of jeans for $35. The tax rate was 6%. What was the total cost of Joseph's purchases?
Responses
$78.96
$89.04
$5.04
$84.00
Answer:
$89.04
Step-by-step explanation:
can yall also pls help me with this one to?
Answer:
Step-by-step explanation:
x > -12
Choose scales for the coordinate plane shown so that you can graph the points J(20, 3), K(25, 3), L(15, −3), M(−5, 5), and N(−5, −4). On the x−axis, use a scale of unit(s) for each grid square. On the y−axis, use a scale of unit(s) for each grid square. Complete the explanation for using using these scales for each axis. The x−coordinates range from to , and the y−coordinates range from _ to _ PLS HURRY!!!
The x-coordinates range from -5 to 25, and the y-coordinates range from -4 to 5.
To choose appropriate scales for the coordinate plane, first, observe the range of x and y coordinates for the given points J(20, 3), K(25, 3), L(15, -3), M(-5, 5), and N(-5, -4).
The x-coordinates range from -5 to 25, and the y-coordinates range from -4 to 5.
For the x-axis, we can use a scale of 5 units for each grid square.
This scale will allow us to cover the entire range of x-coordinates.
The grid will have 6 squares in the positive x-direction and 2 squares in the negative x-direction.
For the y-axis, we can use a scale of 1 unit for each grid square.
This scale will allow us to cover the entire range of y-coordinates.
The grid will have 5 squares in the positive y-direction and 4 squares in the negative y-direction.
In summary, use a scale of 5 units for each grid square on the x-axis, and a scale of 1 unit for each grid square on the y-axis.
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Seth wants to buy a new skateboard that’s costs $167. He has $72. 75 in the bank. If he earns $7. 25 an hour pulling weeds, how many hours will Seth have to work to earn the rest of the money needed to buy the skateboard?
To obtain the remaining funds necessary to purchase the skateboard, Seth must labor for 13 hours at a rate of $7.25 per hour.
To find out how many hours Seth needs to work to earn the remaining money, we need to first calculate how much money he still needs to earn:
Money needed = Cost of skateboard - Money in the bank
Money needed = $167 - $72.75
Money needed = $94.25
Now we can use Seth's hourly rate to figure out how many hours he needs to work to earn the remaining money:
Hours needed to work = Money needed / Hourly rate
Hours needed to work = $94.25 / $7.25 per hour
Hours needed to work = 13 hours (rounded up to the nearest hour)
Therefore, Seth needs to work for 13 hours at $7.25 per hour to earn the remaining money needed to buy the skateboard.
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x²-x-6
2
x+3x² - 2x - 3*
Find a reasonable estimate of the limit lim
According to the given information the limit of the given equation is 1.25 so the correct answer is option a.
What is the definition of a limit in mathematics?Limit, a closeness-based mathematical notion, is largely used to give values to some functions at locations where none are specified, in a manner that is compatible with neighbouring values.
What is meant by "limit of a function"?The value that a function assumes when its input approaches as well as approaches a particular number is really the function's limit. Limits determine continuity, integrals, as well as derivatives. The behaviour of the function at a certain place is always of relevance to the limit of the function.
[tex]\lim_{x \to 3} \frac{x^2-x-6}{x^2 - 2x - 3} \\\\ \lim_{x \to3} \frac{(x-3)(x+2)}{(x-3)(x+1)} \\\\ \lim_{x \to3} \frac{(x+2)}{(x+1)} \\\\$putting the value x=3$\\\\=\frac{3+2}{3+1} \\\\=\frac{5}{4} \\\\=1.25[/tex]
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a bag has 10 blue,12 green,8 yellow,11 pink, and 10 orange pieces of candy. what is the probability of reaching in the bag, without looking, and choosing either a blue or a yellow piece of candy
The probability of reaching into the bag and choosing either a blue or yellow piece of candy is 38/60 or 56%.
What is probability?
The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.
Here, we have
Given: a bag has 10 blue,12 green,8 yellow,11 pink, and 10 orange pieces of candy.
The probability of reaching into the bag and choosing a blue piece of candy is 5/10 or 50%.
The probability of reaching into the bag and choosing a yellow piece of candy is 3/5 or 60%.
Hence, the probability of reaching into the bag and choosing either a blue or yellow piece of candy is 38/60 or 56%.
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PLS HELPP ILL MARK U BRAINLIST and explain ur choice
Answer:
A.
Step-by-step explanation:
Because If you are simplifying then -3 would become -1 and -5 would become -3 and -4 would become -2 so your basically just simplifying
Sam, Keeley and Cody each spun the same spinner a number of times and recorded how many times it landed on a section labelled 2. Their results are shown below. a) They each used their own results to work out the estimated probability of the spinner landing on 2. Which person had the best estimate for the probability? b) By combining all of their results, work out the estimated probability of the spinner landing on 2. Give your answer as a decimal. c) Will using the combined results give a better or worse estimate than using only one person's results? Write a sentence to explain your answer. Number of times the spinner landed on 2 Total number of spins Sam 21 80 Keeley 20 50 Cody 23 70
Comparing the estimated probabilities with the actual probability, we can see that Keeley had the best estimate for the probability.
a) To find out who had the best estimate for the probability of the spinner landing on 2, we need to compare their estimated probabilities with the actual results. The actual probability of the spinner landing on 2 can be found by dividing the total number of times it landed on 2 by the total number of spins.
Sam's estimated probability = 21/80 = 0.2625
Keeley's estimated probability = 20/50 = 0.4
Cody's estimated probability = 23/70 = 0.3286
The actual probability of the spinner landing on 2 = (21+20+23)/(80+50+70) = 0.315
b) To work out the estimated probability of the spinner landing on 2 by combining all of their results, we need to add up the total number of times the spinner landed on 2 and divide it by the total number of spins.
Total number of times the spinner landed on 2 = 21 + 20 + 23 = 64
Total number of spins = 80 + 50 + 70 = 200
Estimated probability of the spinner landing on 2 = 64/200 = 0.32
c) Using the combined results will give a better estimate than using only one person's results. This is because combining the results gives a larger sample size, which makes the estimate more accurate and reliable.
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Karla’s dad is planning to change the fencing around his backyard. The area of the yard and the length is given by (8x+5) ft.is (8x^2+13x+5) ft^2If the fence is along the length and two widths of the yard, determine how many feet of fencing he will need
Answer:
(16x^2 + 46x + 20)/(4x + 5) ft
Step-by-step explanation:
he area of the yard is given by (8x^2+13x+5) ft^2. The length of the yard is (8x+5) ft. The width of the yard is (8x^2+13x+5)/(8x+5) ft.
The fence is along the length and two widths of the yard. Therefore, the length of the fence is 2(8x+5) ft and the width of the fence is 2(8x^2+13x+5)/(8x+5) ft.
The total length of the fence is the sum of the length and width of the fence.
Therefore, the total length of the fence is 2(8x+5) + 2(8x^2+13x+5)/(8x+5) ft.
Simplifying the expression, we get:
2(8x+5) + 2(8x^2+13x+5)/(8x+5) = (16x^2 + 46x + 20)/(4x + 5) ft.
Therefore, Karla’s dad will need (16x^2 + 46x + 20)/(4x + 5) ft of fencing.
which of the following random variables meets the criteria for a hypergeometric distribution? multiple choice question. suppose 30% of the population of have a graduate degree. define x to be the number of adults in a sample of 20 who have earned a graduate degree. the average number of adults who have a graduate degree is 0.7/household. let x be the number of adults in a household who have a graduate degree. out of 50 adults, 10 who have a graduate degree. a sample of 20 is taken. define x to be the number of adults in the sample with a graduate degree.
The following random variable meets the criteria for a hypergeometric distribution: Out of 50 adults, 10 who have a graduate degree. A sample of 20 is taken. Define x to be the number of adults in the sample with a graduate degree.
A hypergeometric distribution is a type of probability distribution that measures the likelihood of a particular number of successes (in this case, adults with a graduate degree) in a sample without replacement.
A random variable that meets the criteria for a hypergeometric distribution is that which meets these conditions: A sample of size n is taken from a population of size N. The population has k successes and N - k failures.
The hypergeometric distribution is used when sampling is done without replacement, where the sample size is small relative to the population size. The sample size is typically less than 10% of the population size. When you have the information of the population, you can use the hypergeometric distribution.
Out of the options provided, the random variable that meets the criteria for a hypergeometric distribution is, Out of 50 adults, 10 who have a graduate degree. A sample of 20 is taken. Define x to be the number of adults in the sample with a graduate degree.
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PLEASE ANSWEERRR!!!
If the dimensions of the triangle in the ingrande quadrupled, what
would be the new perimeter?
Answer:
132 in
Step-by-step explanation:
You want to know the perimeter of a triangle after the dimensions are quadrupled.
Quadrupled"Quadrupled" means "multiplied by 4."
The perimeter is the sum of the lengths of the sides of the triangle:
P = 8 +10 +15 = 33 . . . . inches
If the sides are quadrupled, it is ...
P' = 4(8) +4(10) +4(15) = 4(8 +10 +15) = 4(33) = 132 . . . . inches
As you can see, if the sides are quadrupled, the perimeter is quadrupled.
The new perimeter is 132 inches.
The metro bus traveld 275 miles and used 55 gallons of gas on it route how far does the bus travel in one gallon
The metro bus travels 5 miles per gallon of gas.
The formula used to calculate the distance traveled per gallon of gas is number of miles traveled divided by number of gallons used. Therefore, to calculate how far the metro bus travels in one gallon, we use the following formula:
Distance per gallon = Miles Traveled / Gallons Used
Distance per gallon = 275 miles / 55 gallons
Distance per gallon = 5 miles
Therefore, the metro bus travels 5 miles per gallon of gas.
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The data in the table describes the preferred type of exercise of 9th graders.
Cycling Running Swimming Row Totals
Boy 12% 18% 16%
Girl 14% 21% 19%
Column Totals 100%
Find the marginal relative frequency for students who prefer swimming as their preferred type of exercise.
39%
35%
19%
16%
The frequency for boys is 16% and the frequency for girls is 19%. Adding these two values together gives us the marginal relative frequency of 16%.
The marginal relative frequency of students who prefer swimming as their preferred type of exercise is 16%. Marginal relative frequency is a measure of the proportion of a certain group in relation to the total population. In this case, the marginal relative frequency of students who prefer swimming as their preferred type of exercise is 16%, which means that out of the total population of students, 16% prefer swimming as their preferred type of exercise.To calculate this, we need to add up the frequency of swimming for boys and girls. The frequency for boys is 16% and the frequency for girls is 19%. Adding these two values together gives us the marginal relative frequency of 16%.
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Help me pls I will give you Brainly
Here's the tree diagram for the given scenario:
(1) The probability of getting tails and a 3 is the probability of following the path T-3, which is 1/2 x 1/4 = 1/8.
(ii) The probability of getting heads and an even number is the probability of following the path H-2 or H-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.
(iii) The probability of not getting heads and an even number is the probability of following the paths T-2 or T-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.
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The radius r of a sphere is increasing at a rate of 3 inches per minute. a) Find the rates of changes of the volume when r = 9 inches and r = 36 inches. b) Explain why the rate of change of the volume of the sphere is not constant even though dr/dt is constant.
(a) The rates of changes of the volume are:
When r = 9 inches, dV/dt = 972π cubic inches per minute
When r = 36 inches, dV/dt = 15,552π cubic inches per minute
(b) The rate of change of the volume relates to the radius through a quadratic function not a constant function.
How to find the rates of changes of the volume?a) We know that the formula for the volume of a sphere is V = (4/3)πr³. We can take the derivative with respect to time t to find the rate of change of volume.
dV/dt = 4πr² (dr/dt)
When r = 9 inches, we have:
dV/dt = 4π(9²)(3) = 972π cubic inches per minute
When r = 36 inches, we have:
dV/dt = 4π(36²)(3) = 15,552π cubic inches per minute
b) The rate of change of the volume of the sphere is not constant even though dr/dt is constant because the rate of change of the volume relates to the radius through a quadratic function not a constant function.
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need help with this question
[tex](5x^8y^4)/(15x^3y^8) = (x^5/y^4) * (1/3)[/tex] is the fully simplified expression using only positive exponents.
To simplify [tex](5x^8y^4)/(15x^3y^8)[/tex], we can start by dividing the numerator and denominator by their greatest common factor, which is[tex]5x^3*y^4[/tex]:
[tex](5x^8y^4)/(15x^3y^8) = (x^8/x^3) * (y^4/y^8) * (1/3)[/tex]
Now we can simplify each term separately. First, [tex]x^8[/tex] divided by [tex]x^3[/tex] equals [tex]x^5[/tex], because when we divide two terms with the same base, we subtract their exponents:
[tex]x^8/x^3 = x^(8-3) = x^5[/tex]
Similarly, [tex]y^4[/tex]divided by[tex]y^8[/tex] equals [tex]1/y^4[/tex], because when we divide two terms with the same base, we subtract their exponents and change the sign:
[tex]y^4/y^8 = y^(4-8) = 1/y^4[/tex]
Finally, 1/3 is already in its simplest form. Putting it all together, we get:
[tex](5x^8y^4)/(15x^3y^8) = (x^5/y^4) * (1/3)[/tex]
This is the fully simplified expression using only positive exponents.
We can state that the original equation represents a fraction, where the numerator is the product of 15, the third power of x, and the eighth power of y, and the denominator is the product of 5, the eighth power of x, and the fourth power of y.
We then determined the numerator and denominator's largest common factor, which is 5x*3*y*4, in order to condense the formula. Using the exponentiation principles, we divided the numerator and denominator by this factor before simplifying each term. The end result is a fraction with x raised to the fifth power as the numerator, y raised to the fourth power as the denominator, and 3 as the common factor.
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I NEED HELPPPPPPPPPPPPPPPPPPPPPPPPP
A circular flower bed is 19 m in diameter and has a circular sidewalk around it that is 3 m wide. Find the area of the sidewalk in square meters. Use 3. 14 for pi
Multiplying 13 m by itself and multiplying the result by π gives us the area of the sidewalk as A = 530.94 m². We can calculate the area of the sidewalk with A = πr²
The area of the circular sidewalk around the flower bed can be calculated using the formula for the area of a circle: A = πr². To find the radius of the sidewalk, we need to subtract the radius of the flower bed from the diameter of the flower bed and the sidewalk. Therefore, the radius of the circular sidewalk is (19 m - 3 m - 3 m) = 13 m.Using the formula for the area of a circle, we can calculate the area of the sidewalk with A = πr², where r = 13 m. Substituting 13 m for r, we get A = π(13 m)². Multiplying 13 m by itself and multiplying the result by π gives us the area of the sidewalk as A = 530.94 m².
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real estate ads suggest that 60% of homes for sale have garages, 30% have swimming pools, and 18% have both features. what is the probability that a home for sale has garage but no pool?
The probability of a home for sale having a garage but no pool is 0.42.
To do this, we need to use some probability rules. One useful rule is the formula for calculating the probability of an event not occurring. This is given by:
P(not A) = 1 - P(A)
where A is the event we are interested in, and not A is the event of A not occurring.
In this case, we are interested in the probability of a home having a garage but no pool. Let's call this event GNP (short for garage no pool). Using the information given, we know that 60% of homes have garages and 18% have both garages and pools. We can use this information to calculate the probability of a home having a garage but no pool as follows:
P(GNP) = P(G) - P(G and P)
where G is the event of a home having a garage and P is the event of a home having a pool.
Substituting the values we have:
P(GNP) = 0.6 - 0.18
P(GNP) = 0.42
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an economist wishes to estimate the average family income in a certain population. the population standard deviation is known to be $4,800, and the economist uses a random sample of 225 families. what is the probability that the sample mean will fall within $600 of the population mean?
The probability that the sample mean will fall within $600 of the population of average family income mean is 0.3203 or 32.03%.
Population standard deviation = σ = $4800
Sample size = n = 225
Difference between the sample mean and population mean = d = $600
We need to find the probability that the sample mean will fall within $600 of the population mean.
We can use the z-score formula for sample means to find this probability:
z = (x - μ) / (σ / √n)
where z is the z-score, x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Substituting the given values, we get:
z = ($600) / ($4800 / √225)
z = 0.75
Using the standard normal distribution table or calculator, we can find the probability that z is between -0.75 and 0.75:
P(-0.75 < z < 0.75) = 0.5469 - 0.2266 = 0.3203
Therefore, the probability that the sample mean will fall within $600 of the population of average family income mean is 0.3203 or 32.03%.
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A hydraulic jack lifts an 1162kg automobile a distance of 39cm off the ground, when a force of 6800N pushes on a lever, moving the piston through a distance of 0. 76m. Find the FR, VR, and Efficiency
The FR of the hydraulic jack is 1.94, the VR is 1.95, and the Efficiency is 90%.
To find the FR, VR, and Efficiency of the hydraulic jack, we can use the following formulas:
FR = (F2 / F1)
VR = (D1 / D2)
Efficiency = (F2 x D2) / (F1 x D1)
where F1 is the input force, D1 is the input distance, F2 is the output force, and D2 is the output distance.
Given:
F1 = 6800N
D1 = 0.76m
F2 = ?
D2 = 0.39m
m = 1162kg
To find F2, we can use the formula for work:
Work input = Work output
F1 x D1 = F2 x D2
(6800N) x (0.76m) = F2 x (0.39m)
F2 = (6800N x 0.76m) / 0.39m
F2 = 13,216.41N
To find VR, we can use the formula:
VR = (D1 / D2) = (0.76m / 0.39m) = 1.95
To find Efficiency, we can use the formula:
Efficiency = (F2 x D2) / (F1 x D1)
Efficiency = (13,216.41N x 0.39m) / (6800N x 0.76m)
Efficiency = 0.90 or 90%
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Given the relation of this table:
What is the rule for this relation?
The rule for this relation could be S = n² + 1.
What is relation rule?
A relation rule is a description of how two or more quantities or variables are related to each other. A relation rule can take various forms, depending on the nature of the relation and the variables involved.
For example, a relation rule can be expressed as an equation, a formula, a table, a graph, or a verbal description. In each case, the relation rule specifies the conditions under which the variables are related, and how they vary in relation to each other.
Based on the given table:
n: 1, 0, 2
S: 1, 0, 3
We can observe that when n=1, S=1; when n=0, S=0; and when n=2, S=3.
There are different possible rules that can describe this relation, but one possible rule is:
S = n² + 1
Using this rule, we can verify that it matches the given table:
When n=1, S = 1² + 1 = 2, which matches the given value of S=1.When n=0, S = 0² + 1 = 1, which matches the given value of S=0.When n=2, S = 2² + 1 = 5, which matches the given value of S=3.Therefore, the rule for this relation could be S = n² + 1.
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Pls solve these polynomial
(2x-4)(x+5)
(X-2^2)
(3x+1)^2
(3x-1)(2x^2+5x-4)
1- (2x-4)(x+5)
Multiplying using the distributive property, we get:
(2x-4)(x+5) = 2x(x) + 2x(5) - 4(x) - 4(5)
= 2x^2 + 10x - 4x - 20
= 2x^2 + 6x - 20
Therefore, (2x-4)(x+5) simplifies to 2x^2 + 6x - 20.
2-(x-2)^2
Expanding using the formula for the square of a binomial, we get:
(x-2)^2 = x^2 - 4x + 4
Therefore, (x-2)^2 simplifies to x^2 - 4x + 4.
3- (3x+1)^2
Expanding using the formula for the square of a binomial, we get:
(3x+1)^2 = (3x)^2 + 2(3x)(1) + (1)^2
= 9x^2 + 6x + 1
Therefore, (3x+1)^2 simplifies to 9x^2 + 6x + 1.
4- (3x-1)(2x^2+5x-4)
Using the distributive property, we can multiply each term in the first polynomial by each term in the second polynomial:
(3x-1)(2x^2+5x-4) = 3x(2x^2) + 3x(5x) - 3x(4) - 1(2x^2) - 1(5x) + 1(4)
= 6x^3 + 15x^2 - 12x - 2x^2 - 5x + 4
= 6x^3 + 13x^2 - 17x + 4
Therefore, (3x-1)(2x^2+5x-4) simplifies to 6x^3 + 13x^2 - 17x + 4.
The shaping made from part of circles determine the perimeter
The perimeter of the shape is approximately 314.
The perimeter of a shape made from part of a circle can be found using the formula P = 2πr + 2s, where r is the radius of the circle and s is the length of the straight line segments. For example, if a shape is composed of a semicircle with radius 5 and two straight line segments of length 10, then the perimeter can be calculated as P = 2π(5) + 2(10) = 30π + 20 = 100π. To solve this calculation, we can use the value of π, which is approximately 3.14, to get the approximate perimeter of the shape as 100π ≈ 314.
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Complete question
What is the perimeter of a shape made from part of a circle composed of a semicircle with radius 5 and two straight line segments of length 10?
One third of a number is 3054. What is the whole number?
Step-by-step explanation:
let the number be y
one-third of the number = 1/3 × y = y/3
the total is 3054
therefore,. y/3 = 3054
multiply both sides by 3
y/3 × 3 = 3054 × 3
y = 9162
Thus, the whole number is 9162
3054 is 1/3 of a whole number.
In order to find the whole number, you need to find the other 2/3's. Since we already have the 1/3, we can easily solve this problem.
3054x3 = 9162
3054+3054+3054 = 9162
The whole number is 9162.
Hope this helped!
I need help with this two !!!!!!
Answer:
#15 (a)
[tex]{ \sf{a + 2b - c + 4d}} \\ \\ \hookrightarrow \: { \sf{ \binom{4}{6} + 2 \binom{4}{ - 2} - \binom{ - 3}{ - 2} + 4 \binom{5}{1} }} \\ \\ \hookrightarrow \: { \sf{ \binom{4}{6} + \binom{8}{ - 4} + \binom{3}{2} + \binom{20}{4} \: \: \: \: \: \: }} \\ \\ { \sf{ \hookrightarrow \: \binom{4 + 8 + 3 + 20}{6 - 4 + 2 + 4} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \sf{ \hookrightarrow \: { \boxed{ \binom{45}{8} }}}}[/tex]
#15 (b)
Step-by-step explanation:
15 (a)
if it is a vector
First, we need to multiply each vector component by its corresponding scalar value and then add them together.
a + 2b - c + 4d = (4)(1) + (4)(2) + (-3)(-1) + (5)(4) = 4 + 8 + 3 + 20 = 35
Therefore, the value of a + 2b - c + 4d is 35.
if it is a matrix
we can form a 1x4 matrix for vector a, b, c, and d respectively. We can also form a 4x4 matrix for the scalar values. Then we can perform matrix multiplication as follows:
(4 4 -3 5) (1 0 0 0) (0 2 0 0) (0 0 -1 0) (0 0 0 4)
= (41 + 40 - 30 + 50) (40 + 42 - 30 + 50) (40 + 40 - 3*(-1) + 50) (40 + 40 - 30 + 5*4)
= (4, 8, 3, 20)
Therefore, the value of a + 2b - c + 4d is (4, 8, 3, 20).
15 (b)
To find the product of the two matrices, we need to multiply each element of the first matrix by its corresponding element in the second matrix and add the products. The resulting matrix will be a 2 x 2 matrix.
| 5 -2 | | 2 6 | |(5)(2) + (-2)(5) (5)(6) + (-2)(3)|
| 4 1 | x | 5 3 | = |(4)(2) + (1)(5) (4)(6) + (1)(3)|
Performing the matrix multiplication, we get:
| 5 -2 | | 2 6 | | 10 -12 |
| 4 1 | x | 5 3 | = | 13 27 |
Therefore, the product of the matrices (5 -2, 4 1) and (2 6, 5 3) is the 2 x 2 matrix:
| 10 -12 |
| 13 27 |
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if a snowball melts so that its surface area decreases at a rate of 2 cm2/min, find the rate at which the diameter decreases when the diameter is 11 cm.
The rate at which the diameter decreases when the diameter is 11 cm is 4 cm/min.
To solve this question, we need to use the formula for the surface area of a sphere, A = 4πr2. Since the surface area of the snowball is decreasing at a rate of 2 cm2/min, we can set up a differential equation to calculate the rate of change of the radius with respect to time: dA/dt = 2 cm2/min = 8π dr/dt.
We can rearrange this equation to find dr/dt, the rate at which the radius of the snowball is changing: dr/dt = 2 cm2/(8πmin).
We then use the equation for the diameter of a circle, d = 2r, and substitute the rate of change of the radius to get the rate of change of the diameter: dd/dt = 2(dr/dt) = 2 cm2/(4πmin).
Finally, we can plug in the given diameter, 11 cm, to find the rate of decrease in diameter: dd/dt = 11 cm/(4πmin) = 4 cm/min. Therefore, the rate at which the diameter decreases when the diameter is 11 cm is 4 cm/min.
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A circular pond is 21 m in diameter. It is surrounded by 3.5 m wide path. Find the
cost of constructing the path at the rate of Rs. 25 per m2.
The cost of constructing the path at the rate of Rs. 25 per m2 would be Rs. 38,465.
To find the cost of constructing the path, we need to determine the area of the path and then multiply it by the cost per square meter.
The total diameter of the pond and the path is 21m + 3.5m + 3.5m = 28m.
So the radius of the pond is half of the diameter, which is 21m/2 = 10.5m.
The radius of the pond with the path is (21m+3.5m)/2 = 12.25m.
Area of the path = Area of the outer circle - Area of the pond
[tex]= π(12.25)^2 - π(10.5)^2[/tex]
[tex]= π[(12.25)^2 - (10.5)^2][/tex]
= 3.14 x (24.5 + 10.5) x (24.5 - 10.5)
= 3.14 x 35 x 14
= 1538.6 m2 (rounded to one decimal place)
Therefore, the cost of constructing the path at the rate of Rs. 25 per m2 would be:
Cost = Area x Rate per m2
= 1538.6 x 25
= Rs. 38,465.
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Can someone please explain and answer this for me I’m so lost
Answer:
[tex]\left(\begin{array}{cc}0&24\\13&27\end{array}\right)[/tex]
Step-by-step explanation:
I will attempt to explain using the following example. Let us consider the product (multiplication) of the following matrices:
[tex]\left(\begin{array}{cc}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\end{array}\right) \cdot \left(\begin{array}{cc}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\end{array}\right) = \left(\begin{array}{cc}c_{1,1}=c_{1\times 1}&c_{1,2}=c_{1\times2}\\c_{2,1}=c_{2\times1}&c_{1,2}=c_{2\times2}\end{array}\right)[/tex]
Note that the first number in each coefficient refers to the row number, while the second number refers to the column number.
Then [tex]a_{2,1}[/tex] indicates the element in the second row and first column.
To calculate the product of two matrices, we need to multiply each row of the first matrix by each column of the second matrix.
Example:
[tex]c_{1,1}= (a_{1,1} \cdot b_{1,1} + a_{1,2} \cdot b_{2,1})[/tex]
Now, we can apply this to the original exercise:
[tex]\left(\begin{array}{cc}5&-2\\4&1\end{array}\right) \left(\begin{array}{cc}2&6\\5&3\end{array}\right) =\left(\begin{array}{cc}c_{1,1}&c_{1,2}\\c_{2,1}&c_{2,2}\end{array}\right)[/tex]
Next, we will calculate each value of c using the multiplication process we just discussed.
[tex]c_{1,1}= (5(2)+-2(5)) = 10-10 =0\\c_{1,2}= (5(6)+-2(3)) = 30-6 =24\\c_{2,1}= (4(2)+1(5)) = 8+5 =13\\c_{2,2}= (4(6)+1(3)) = 24+3 =27\\[/tex]
Thus, we have obtained the final result of the matrix product:
[tex]\left(\begin{array}{cc}5&-2\\4&1\end{array}\right) \left(\begin{array}{cc}2&6\\5&3\end{array}\right) =\left(\begin{array}{cc}0&24\\13&27\end{array}\right)[/tex]
[tex]\text{-B$\mathfrak{randon}$VN}[/tex]