The correct answer is (D) 132.7 feet.
From the given information, we can draw a diagram as follows:
A
/|
/ |
/ | 37°
/ |
/ |
/ |
/ |
/_ _ _ _|
d B
Where A represents the top of the cliff, B represents the ship and d represents the distance between the ship and the bottom of the cliff.
Since we know that the angle of depression is 37 degrees, then the angle CAB is also 37 degrees. We also know that AB is 100 feet, which is the height of the cliff. We want to find the distance d between the bottom of the cliff and the ship.
We can use the tangent function to find d:
tan(37°) = AB/BD
tan(37°) = 100/d
d = 100/tan(37°)
Using a calculator, we can find that:
d ≈ 132.7 feet
Therefore, the correct answer is (D) 132.7 feet.
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A home repair crew charges $75 dollars per day plus a $250 service fee. the total amount the home repair crew charges this client is $925.how many days did the crew work?
The home repair crew worked for 9 days.
What is the duration of the home repair crew's work to charge $925?To determine the number of days the home repair crew worked for, we can use algebra. Let's assume that the number of days they worked for is represented by "d". We know that they charge $75 per day plus a $250 service fee, so we can set up the following equation:
75d + 250 = 925
Simplifying the equation, we get:
75d = 675
Dividing both sides by 75, we get:
d = 9
However, we need to keep in mind that the $250 service fee is a one-time charge, not a daily charge. So we need to subtract that from the total amount to get the actual amount charged for the days worked:
925 - 250 = 675
Dividing 675 by the daily rate of $75, we get:
675 / 75 = 9
Therefore, the home repair crew worked for 9 days.
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A triangle has angle measures of (x + 3)°, (5x – 8)°, and (2x + 1)°.
What is the measure of the smallest angle of the triangle in degrees?
A 47°
B 26°
C 107°
D 23°
Answer: B
Step-by-step explanation:
x + 3 + 5x - 8 + 2x + 1 = 180
8x - 4 = 180
8x = 184
x = 23
23 + 3 = 26, 5(23) - 8 = 107, 2(23) + 1 = 47
the smallest is 26
the picture below is being enlarged by a scale factor of 2.5. how many inches of farming will the picture require? base 5 in high4 in
A. 12.5 in
B. 20 in
C. 45 in
D. 65 in
The inches of farming the picture require is 45 inches
How many inches of farming will the picture require?From the question, we have the following parameters that can be used in our computation:
Base = 5 in
High = 4 in
Scale factor = 2.5
The inches of farming the picture require is calculated as
Perimeter = 2 * (Base + High) * Scale factor
Substitute the known values in the above equation, so, we have the following representation
Perimeter = 2 * (5 + 4) * 2.5
Evaluate
Perimeter = 45
Hence, the inches of farming the picture require 45 inches
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Select all ordered pairs that satisfy the function y=-4x+20
6,4
0,20
-4,20
10,-20
The ordered pairs that satisfy the function is B)(0,20) and D)(10,-20).
What is function?
A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
Here the given function is y=-4x+20.
Now put x=6 and y=4 then,
=> 4=-4(6)+20
=> 4 = -24+20
=> 4 ≠ -4.
Then the coordinate (6,4) dost not satisfy the function.
Put x=0 and y=20 then,
=> 20 = -4(0)+20
=> 20= 0+20
=> 20=20
Hence the coordinate (0,20) satisfy the function.
Now put x=-4 and y=20 then,
=> 20 = -4(-4)+20
=> 20 = 16+20
=> 20 ≠ 36
Hence the coordinate (-4,20) does not satisfy the function.
Now put x=10 and y=-20 then,
=> -20 = -4(10)+20
=> -20 = -40+20
=> -20=-20
Then the coordinate (10,-20) satisfy the function.
Hence the ordered pairs that satisfy the function is B)(0,20) and D)(10,-20).
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PLEASE HELP MEEE!!! WILL GIVE BRAINLIEST!!!
Answer:
75°
Step-by-step explanation:
Arc length LM is 108°, meaning that angle <K is half of the arc length, 54°.
*REMEMBER* all 3 angles inside a triangle must add up to 180° to make it a triangle.
We have angle <L, <K, and we are looking for angle <M.
51+54+?=180
105+?=180
?=75
Angle <M is 75°
Hope this helps :)
Mathematical
PRACTICE
Use Algebra For Exercises 11-13,
2 x P
40
refer to the equation 5 x 9
100
2
11. What must be true about p and q if the equation show
equivalent fractions?
p and q both are equal and p=q=20.
Given are an equation show equivalent fractions 2p/5q = 40/100
We need to find the p and q,
So
2p/5q = 40/100
2p/5q = 2×20/2×20
p/q = 20/20
p/q = 1
p = q
Therefore p and q both are equal and p=q=20.
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Use the drop down to answer the question about converting 0. 64 to a fraction.
How many repeating digits are in 0. 64 ?
What value is multiplied on both sides of the equals sign?
What fraction represents 0. 64 ?
The fraction value of 0.64 is 16/25 using the greatest common factor method to represent. There are no repeating digits in value.
The given decimal value = 0.64
There are no repeating numbers in the given decimal number 0.64.
To convert the given number into a fraction, we need to multiply the value with 10 on both sides to move the decimal point two places to the right.
0.64 × 100/100 = 64/100
Now simply this value by dividing both the numerator and denominator with the greatest common factor of both values. The greatest common factor is 4.
64/100 = 16/25
Therefore, we can represent the fraction value of 0.64 as 16/25.
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Map of the city zoo a triangle with points zebras, monkeys, and lions. the distance from zebras to monkeys is 52 feet and from monkeys to lions is x feet. a triangle with points lions, tigers, elephants. the distance from lions to tigers is 96 feet and from tigers to elephants is 78 feet. the path from the zebras to the monkeys is parallel to the path from the tigers to the elephants. what is the distance between the lions and monkeys? 1. proportion: 52 78 = x 96 2. cross-multiply: 4992 = 78x 3. solve: the distance between the lions and the monkeys is feet.
The distance between the lions and the monkeys is 64 feet.
We can set up a proportion to find the distance between the lions and monkeys. Here's the step-by-step explanation:
1. Proportion: Since the path from zebras to monkeys is parallel to the path from tigers to elephants, we can set up a proportion using the given distances: 52/78 = x/96.
2. Cross-multiply: To solve for x, we can cross-multiply: 52 * 96 = 78 * x, which simplifies to 4992 = 78x.
3. Solve: Now we just need to solve for x. Divide both sides of the equation by 78: x = 4992 / 78. This gives x ≈ 64.
So, the distance between the lions and the monkeys is approximately 64 feet.
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Zelda has 8 rabbits with which to start an animal farm. If the rabbit population doubles each month, in how many months will the rabbit population be 5,800?
In a case whereby Zelda has 8 rabbits with which to start an animal farm. If the rabbit population doubles each month, the number of months that the rabbit population will be 5,800 is 9.5 months.
How can the the number of months?In order to calculatre the month then we can use the expression y = abⁿ
a = starting number ( 8 rabbits)
b = rate of change = (2)
n = number of months that we need to calculate
y = rabbit population = 5800
The we can substitute to have
5800 = 8 × 2ⁿ
5800 / 8 = 2ⁿ
725 = 2ⁿ
n = 9.5 months.
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The owner of a meat market has an assistant who has determined that the weights of roasts are normally distributed, with a mean of 3.2 pounds and standard deviation of 0.8 pounds. for a sample of 25 roasts, what is the probability of sample mean less than 3.1 pounds
The probability of sample mean less than 3.1 pounds: 0.266. The correct option is C.
The weights of roasts are normally distributed with a mean of 3.2 pounds and a standard deviation of 0.8 pounds. In a sample of 25 roasts, we want to find the probability that the sample mean is less than 3.1 pounds.
First, we need to calculate the standard error of the mean, which is the standard deviation divided by the square root of the sample size:
Standard error = 0.8 / √25 = 0.8 / 5 = 0.16
Now, we need to calculate the z-score for the sample mean of 3.1 pounds:
Z = (Sample mean - Population mean) / Standard error = (3.1 - 3.2) / 0.16 = -0.1 / 0.16 = -0.625
Using a z-table or calculator, we find the probability associated with a z-score of -0.625:
P(Z < -0.625) ≈ 0.266
Therefore, the probability of the sample mean being less than 3.1 pounds is approximately 0.266, which corresponds to option C.
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Complete question:
The owner of a meat market has an assistant who has determined that the weights of roasts are normally distributed, with a mean of 3.2 pounds and a standard deviation of 0.8 pounds. For a sample of 25 roasts, what is the probability of sample mean less than 3.1 pounds?
a. 0.495
b. 0.450
c. 0.266
d. 0.521
percent of birds in Forest A that are robins
Answer: To determine the percent of birds in Forest A that are robins, we need to know the number of robins and the total number of birds in Forest A. Let's assume that we have conducted a bird survey and found the following information:
Number of robins in Forest A = 50
Number of other birds in Forest A = 150
Total number of birds in Forest A = 50 + 150 = 200
To calculate the percentage of birds in Forest A that are robins, we can use the following formula:
Percentage = (Number of robins / Total number of birds) x 100%
Plugging in the values, we get:
Percentage = (50 / 200) x 100% = 25%
Therefore, 25% of the birds in Forest A are robins.
Step-by-step explanation:
To find the quotient of 4. 082 and 10,000, move the decimal point in 4. 082
Choose. Right left
places to the
Choose. Right-left
The quotient of 4.082 and 10,000 is 0.000004082.
To find the quotient of 4.082 and 10,000, we need to divide 4.082 by 10,000. However, dividing a decimal by another decimal can be tricky, so we need to move the decimal point of the dividend (4.082) and the divisor (10,000) to make the division easier.
We can move the decimal point of 4.082 four places to the left to obtain 0.04082. This is because moving the decimal point to the left makes the number smaller. For example, moving the decimal point in 4.082 one place to the left gives us 0.4082, which is ten times smaller than 4.082. Moving the decimal point four places to the left gives us a number that is 10,000 times smaller than 4.082.
Similarly, we can move the decimal point of 10,000 four places to the right to obtain 100,000,000. This is because moving the decimal point to the right makes the number larger. For example, moving the decimal point in 10,000 one place to the right gives us 1,000, which is ten times larger than 10,000. Moving the decimal point four places to the right gives us a number that is 10,000 times larger than 10,000.
Now we can divide 0.04082 by 100,000,000, which gives us the quotient of 0.000004082. Therefore, to find the quotient of 4.082 and 10,000, we need to move the decimal point in 4.082 four places to the left and move the decimal point in 10,000 four places to the right, and then divide the resulting numbers.
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a bank account earns 0.06% annual interest compounded monthly the bank B of the account after T months started with the $250 is given by the equation B =250(1.06)^t. How long will it take to triple the balance of the account?
It will take about 387.3 months (or about 32.3 years) to triple the balance of the account.
What is Compound Interest ?
Compound interest refers to the process of earning interest on both the initial principal amount as well as any accumulated interest from previous periods. In other words, it is the interest that is earned on the interest that has been accumulated over time.
We can use the formula for compound interest to solve this problem.
Where:
A = the final amount
P = the initial amount
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time (in years)
In this problem, we have:
P = $250
r = 0.06% = 0.0006 (as a decimal)
n = 12 (since the interest is compounded monthly)
A = 3P = $750
Substituting these values into the formula, we get:
750 = 250[tex](1 + 0.0006/12) ^{12t}[/tex]
Dividing both sides by 250, we get:
3 =[tex](1 + 0.0006/12) ^{12t}[/tex]
Taking the natural logarithm of both sides, we get:
㏒(3) = 12t ㏒(1 + 0.0006÷12)
Solving for t, we get:
t = ㏒(3)/(12 ㏒(1 + 0.0006÷12))
Plugging this into a calculator, we get:
t ≈ 387.3 months
Therefore, it will take about 387.3 months (or about 32.3 years) to triple the balance of the account.
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please help.
1. Complete the Pythagorean triple. (24,143, ___)
2. Given the Pythagorean triple (5,12,13) find x and y
3. Given x=10 and y=6 find associated Pythagorean triple
4. Is the following a possible Pythagorean triple? (17,23,35)
The value that will complete Pythagorean triple would be = 24,143, 145 )
How to calculate the missing value of a triangle using the Pythagorean formula?To calculate the missing value of a triangle that completes a Pythagorean triple that formula that should be used is given as follows.
That is;
C ² = a² + b²
C = Missing value of the Pythagorean triple
a = 24
b.= 143
C² = 24²+143²
= 576+20,449
C =√21,025
= 125
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Charlene sees a laptop being sold for $492. this is 18% less than the original price. what was the original price of the laptop?
The original price of the laptop was $600 because $492 is 82% of the original price (100% - 18% = 82%).
How to find the percentage of the laptop?The concept of percentage decrease. A percent decrease is the amount by which a quantity decreases, expressed as a percentage of its original value. In this case, the laptop is being sold for 18% less than its original price, so we can represent the percent decrease as 18%.
To find the original price of the laptop, we need to work backwards from the sale price. We can use the formula:
Sale price = Original price - Percent decrease of original price
where "Percent decrease of original price" is the percentage decrease of the original price, expressed as a decimal. In this case, the percent decrease is 18%, which we convert to a decimal by dividing by 100: 18/100 = 0.18.
Plugging in the values we know, we get:
$492 = Original price - 0.18 * Original price
Simplifying this equation, we get:
$492 = 0.82 * Original price
To isolate Original price, we can divide both sides by 0.82:
Original price = $492 / 0.82
Simplifying this expression, we get:
Original price = $600
the original price of the laptop was $600.
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Question 16 (6 marks) If b and c are real numbers and b^2 <3c, show that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution.
We have shown that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution.
To show that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution, we will use the discriminant (∆) of the equation. The discriminant helps us determine the nature of the solutions of a polynomial equation.
For a cubic equation, the discriminant is given by the following formula:
∆ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2
In our case, the equation is x^3 + bx^2 + cx - 2022 = 0, so a = 1, b = b, c = c, and d = -2022.
Now, let's calculate the discriminant:
∆ = 18(1)(b)(c)(-2022) - 4b^3(-2022) + b^2c^2 - 4(1)c^3 - 27(1)^2(-2022)^2
∆ = -36444bc + 8088b^3 + b^2c^2 - 4c^3 - 109222392
We are given that b^2 < 3c. This inequality implies that the first three terms of the discriminant will be negative, as b^2c^2 will be smaller than 3c^2. The negative terms will dominate the discriminant, making ∆ < 0.
When the discriminant of a cubic equation is negative (∆ < 0), it means that the equation has exactly one real solution. Thus, we have shown that the equation x^3 + bx^2 + cx = 2022 has exactly one real solution.
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Answer Immeditely Please
Answer:
it's ither 1 2 or 4 I'm still thinking but I'm pretty sure it's one of those answers
A wildlife refuge in South America has howler monkeys and spider monkeys. A biologist working there randomly selected eight adults of each type of monkey, weighed them, and recorded their weights in pounds. Show your work.
howler monkey: {16, 17, 18, 18, 18, 20, 22, 23}
spider monkey: {8, 10, 10, 11, 11, 12, 14, 14}
Calculate the meaning and MAD for each type of monkey.
Calculate the means-to-MAD ratio for the two types of monkeys.
What inference can be made about the weight of both types of monkeys? Explain.
Howler monkeys are heavier on average than spider monkeys (17.75 lbs vs. 11.25 lbs).
How to solveCalculate the mean:
Some of the weights divided by the number of monkeys
Calculate the MAD:
Find the absolute deviation (difference) of each weight from the mean
Calculate the average of these deviations
Mean Howler Monkey = (16+17+18+18+18+20+22+23)/8 = 142/8 = 17.75 lbs
Mean Spider Monkey = (8+10+10+11+11+12+14+14)/8 = 90/8 = 11.25 lbs
Now, we calculate the MAD for each type of monkey:
MAD Howler Monkey= 1.875 lbs
MAD Spider Monkey= 1.5625 lbs
we calculate the means-to-MAD ratio for both types of monkeys:
Howler Monkey: Mean/MAD = 17.75/1.875 = 9.466
Spider Monkey: Mean/MAD = 11.25/1.5625 = 7.2
Inference:
Howler monkeys are heavier on average than spider monkeys (17.75 lbs vs. 11.25 lbs).
The means-to-MAD ratio shows that howler monkeys have more consistent weights (9.466) compared to spider monkeys (7.2), as a higher ratio indicates less variability in weights.
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Doug is filing singly. his net taxable income is $80,575. every week, $304 is withheld from his earnings for income tax. based on the table below, what can doug expect when his taxes are due? between 80,550 and 80,600 dollars, for filing single, the amount of taxes is 16,539 dollars. a. doug will receive a refund of $123. b. doug will receive a refund of $2,977. c. doug will owe an additional $1,125. d. doug will owe an additional $731.
Doug is filing singly, and his net taxable income is $80,575. The tax amount for this income range is $16,539. Every week, $304 is withheld from his earnings for income tax. Doug can expect he will owe an additional $731. So option d is the correct answer.
Calculate the total amount withheld for the year.Since the result is a positive number, Doug will owe an additional $731 when his taxes are due. Therefore, the correct answer is d. Doug will owe an additional $731.
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I need this answer asap!!! Willa can cover 13. 5m squared with 3L of paint and complete the table using equivalent ratios.
13. 5sqaured to 3
? to1
? to 10
The completed table of equivalent ratios would be:
13.5 square meters : 3 liters of paint
4.5 square meters : 1 liter of paint
45 square meters : 10 liters of paint.
To find the missing ratios, we need to set up proportions using the given information.
13.5 square meters is covered by 3 liters of paint, so we can write:
13.5/3 = ?/1
To solve for the missing ratio, we can cross-multiply:
13.5 x 1 = 3 x ?
? = (13.5 x 1) / 3
? = 4.5
So, 4.5 square meters can be covered by 1 liter of paint.
To find the last missing ratio, we can use the same method:
13.5/3 = ?/10
10 x 13.5 = 3 x ?
? = (10 x 13.5) / 3
? = 45
So, 45 square meters can be covered by 10 liters of paint.
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Can someone help me asap? It’s due today!! Show work! I will give brainliest if it’s correct and has work
Answer:
10 outcomes
Step-by-step explanation:
if 2 coins were selected with replacement=10×10=100
number of outcomes if 2 coins were selected without replacement=10×9=90
Finally, 100-90= 10 outcomes!
bruce's weigh 11 each if bruce has 14 how many pounds do they weigh all together
All the Bruces together weigh a total of 154 pounds.
If each Bruce weighs 11 pounds and there are 14 Bruces, we can calculate the total weight by multiplying the weight of each Bruce by the number of Bruces.
Weight of each Bruce: 11 pounds
Number of Bruces: 14
To find the total weight, we multiply 11 pounds by 14 Bruces:
Total weight = 11 pounds/Bruce * 14 Bruces
= 154 pounds
Therefore, all the Bruces together weigh a total of 154 pounds.
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Intelligence Quotient (IQ) scores are often reported to be normally distributed with μ=100. 0 and σ=15. 0. A random sample of 45 people is taken. Step 1 of 2 : What is the probability of a random person on the street having an IQ score of less than 96? Round your answer to 4 decimal places, if necessary
We are given that IQ scores are normally distributed with mean μ = 100 and standard deviation σ = 15. We want to find the probability of a random person on the street having an IQ score of less than 96.
To do this, we need to standardize the IQ score using the z-score formula:
z = (x - μ) / σ
where x is the IQ score we're interested in, μ is the mean IQ score, and σ is the standard deviation of IQ scores.
Plugging in the given values, we get:
z = (96 - 100) / 15 = -0.267
Now, we look up the probability of getting a z-score less than -0.267 in a standard normal distribution table or using a calculator. The probability is approximately 0.3944.
Therefore, the probability of a random person on the street having an IQ score of less than 96 is 0.3944 (rounded to 4 decimal places).
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Noel borrows $800
and is charged compound interest at 60%
per year. How much will he have to pay back in total after 8
years?
Answer:
Step-by-step explanation:
Alexandra likes to skate at a game store that is due south of her school and due west of her
favorite skate park. If the game store is 7.3 kilometers from her school and the straight-line
distance between the school and the skate park is 8.7 kilometers, how far is the game store
from the skate park? If necessary, round to the nearest tenth.
We can use the Pythagorean theorem to find the distance between the game store and the skate park.
Let's call the distance between the game store and the skate park "d". Then we have:
d^2 = (distance between school and game store)^2 + (distance between school and skate park)^2
d^2 = 7.3^2 + 8.7^2
d^2 = 106.18
d ≈ 10.3
Therefore, the game store is approximately 10.3 kilometers from the skate park.
Need help asap.
(Angles Formed by Chords, Tangents, Secants)
The value of angle LQ in the intersected arc is 34 degrees.
How to find arc angle when secant intersect?If two secants intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs.
Therefore, let's find the arc angle LQ as follows:
let
x = arc angle LQ
m∠U = 1 / 2 (108 - x)
37 = 54 - 0.5x
subtract 54 from both sides of the equation
37 = 54 - 0.5x
37 - 54 = 54 - 54 - 0.5x
-17 = -0.5x
divide both sides by -0.5
x = -17 / -0.5
x = 34 degrees
Therefore,
arc angle LQ = 34 degrees
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Mr linden drives from his home to his office every day. if he drives at an average speed of 70 km/h for 45 min, what is the distance of the journey from his home to his office?
If he drives at an average speed of 70 km/h for 45 min, the distance of the journey from Mr. Linden's home to his office is 52.5 km.
To find the distance of Mr. Linden's journey from his home to his office, we can use the formula:
Distance = Speed x Time
Since Mr. Linden drives at an average speed of 70 km/h for 45 minutes, we first need to convert the time to hours:
45 minutes = 0.75 hours
Now, we can plug in the values we have into the formula:
Distance = 70 km/h x 0.75 hours
Distance = 52.5 km
Therefore, the distance of the journey from Mr. Linden's home to his office is 52.5 km.
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7) A tree casts a shadow that is 10 feet long. 5 foot woman standing nearby casts a shadow that is 4 feet long. How tall is the tree?
Answer:
12.5 ft
Step-by-step explanation:
x/10=5/4
4x=5(10)
4x=50
x=12.5
The table shows the daily amount that Trevor spent on snacks. During
which week did Trevor spend a mean amount of $0. 85 per day on snacks? *
1 point
Week
Friday
$0. 50
1
Monday Tuesday Wednesday Thursday
$0. 75 $0. 50
$1. 00 $1. 25
$1. 25 $0. 75 $0. 25 $1. 00
$0. 50 $0. 75
$0. 25 $0. 25
$1. 25 $0. 25 $0. 75 $1. 00
$1. 00
AWN
$1. 25
$0. 50
Week 1
Week 2
Week 3
Week 4
Trevor spent a mean amount of $0.85 per day on snacks during
Week 1.
We have,
To determine during which week Trevor spent a mean amount of $0.85 per day on snacks, we need to calculate the mean (average) amount spent per day for each week and find the week that matches $0.85.
Let's calculate the mean amount spent per day for each week:
Week 1:
= (0.50 + 0.75 + 0.50 + 1.00 + 1.25 + 1.25 + 0.75 + 0.25 + 1.00) / 9
= $0.86 (approximately)
Week 2:
= (1.00 + 0.50 + 0.75 + 0.25 + 0.25 + 1.25 + 0.25) / 7
= $0.61 (approximately)
Week 3:
= (0.75 + 1.00) / 2
= $0.88
Week 4:
= (1.25 + 1.00 + 1.00 + 1.25 + 0.50) / 5
= $1.00
From the calculations, we can see that the mean amount spent per day for Week 1 is closest to $0.85.
Therefore,
Trevor spent a mean amount of $0.85 per day on snacks during
Week 1.
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Find the absolute maximum and minimum values of f(x,y) = x^2 + 2y^2 on the square [0,1] x [0,1]
Answer:
The maximum is when both X and y = 1.The maximum value of the function is 3.When both X and YY are equal to 0, the minimum value is 0
The absolute maximum value is 3 and the absolute minimum value is 0.
To find the absolute maximum and minimum values of f(x,y) = x^2 + 2y^2 on the square [0,1] x [0,1], we need to consider both the interior and the boundary of the square.
First, check for critical points in the interior by finding the partial derivatives and setting them equal to zero:
fx = 2x and fy = 4y
Setting them equal to zero, we have:
2x = 0 => x = 0
4y = 0 => y = 0
The only critical point in the interior is (0,0).
Next, evaluate f(x,y) on the boundary of the square [0,1] x [0,1]. The boundary consists of four segments: x=0, x=1, y=0, and y=1.
1. x=0: f(0,y) = 2y^2 (for y in [0,1])
2. x=1: f(1,y) = 1 + 2y^2 (for y in [0,1])
3. y=0: f(x,0) = x^2 (for x in [0,1])
4. y=1: f(x,1) = x^2 + 2 (for x in [0,1])
Now, compare the values of f at the critical point and boundary points to find the absolute maximum and minimum:
Absolute minimum: f(0,0) = 0
Absolute maximum: f(1,1) = 3
So the absolute maximum value is 3 and the absolute minimum value is 0.
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