The equation has at least one solution in the interval (-1, 1).
To prove that the equation has at least one solution in the interval (-1, 1), we can use the Intermediate Value Theorem.
First, let's simplify the equation by finding a common denominator:
a(x^3+x-2) + b(x^3+2x^2-1) = 0
Now, let's define a new function f(x) = a(x^3+x-2) + b(x^3+2x^2-1). This function is continuous on the interval (-1, 1) because it is a sum of continuous functions.
Next, we will evaluate f(-1) and f(1) to see if the Intermediate Value Theorem can be applied.
f(-1) = a(-1^3-1-2) + b(-1^3+2(-1)^2-1) = -a-b < 0
f(1) = a(1^3+1-2) + b(1^3+2(1)^2-1) = a+3b > 0
Since f(-1) is negative and f(1) is positive, there must be at least one value of x in the interval (-1, 1) such that f(x) = 0, by the Intermediate Value Theorem.
To prove that the given equation has at least one solution in the interval (-1, 1), we can use the Intermediate Value Theorem (IVT). Let's define the function f(x) as follows:
f(x) = a/(x^3 + 2x^2 - 1) + b/(x^3 + x - 2)
Since a and b are positive numbers, we can examine the behavior of f(x) at the endpoints of the interval (-1, 1).
f(-1) = a/((-1)^3 + 2(-1)^2 - 1) + b/((-1)^3 + (-1) - 2)
f(-1) = a/(-1) + b/(-4) < 0
f(1) = a/(1^3 + 2(1)^2 - 1) + b/(1^3 + 1 - 2)
f(1) = a/(2) + b/(0) = a/2 > 0
Since f(-1) < 0 and f(1) > 0, by the Intermediate Value Theorem, there must be at least one point c within the interval (-1, 1) where f(c) = 0. This means that the given equation has at least one solution in the interval (-1, 1).
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i need this answer in by 6:00.. i have tutoring at that time
Answer:
80
Step-by-step explanation:
v=bxh
v=10x8
v=80
Luis created a spreadsheet of his expenses for three months. Which of Luis's expenses are variable expenses
utility bill
Expenses Jan Feb Mar
rent
$1,250,00 $1,250,00 $1,250,00
$124. 11 $108. 72 $121. 69
car loan payment $384. 00 3384,00 $384. 00
Insurance payment 397. 18 597. 18 $97. 18
groceries
$315,43 $367. 25 $341. 04
clothing
$72. 18 $152. 74 $0. 00
fuel
$108. 71 $117. 46 $127. 34
Variable expenses are expenses that fluctuate from month to month, and are typically not a fixed amount. Examples of variable expenses include groceries, fuel, clothing, and entertainment. These expenses can be influenced by various factors such as personal choices, seasonality, and external events.
In Luis's expenses, the following expenses are variable expenses:
Groceries: The amount spent on groceries changes from month to month depending on the types of food Luis purchases and the quantity he buys.Clothing: This expense is variable because Luis only spent money on clothing in January and February, and did not spend anything on clothing in March.Fuel: The amount spent on fuel changes from month to month depending on how often Luis drives and the price of gasoline.On the other hand, the following expenses are fixed expenses:
Rent: This expense is fixed because Luis pays the same amount for rent every month.Car loan payment: This expense is fixed because Luis is required to pay the same amount for his car loan every month.Insurance payment: This expense is fixed because Luis is required to pay the same amount for his insurance every month.Utility bill: The utility bill could be either a variable or fixed expense, depending on the type of utility. For example, if the utility bill is for electricity, it may be a variable expense because the amount of electricity used can fluctuate from month to month. However, if the utility bill is for a fixed service such as internet, it would be a fixed expense.To learn more about “utility” refer to the https://brainly.com/question/14729557
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Answer:
groccieries is the answer for plato 2023
What is the explicit formula for this sequence?
-7, -3, 1, 5, ...
A. an = 9+ (n − 1)(-4)
B. an-4+ (n-1)(-7)
C. an-7+ (n − 1)4
D. an-7+ (n-1)(-4)
Answer:
It is C
Step-by-step explanation:
It cannot be d bc - 7-4=-11 so rejected
Not b bc the first number is - 7
The same w a
question subtract. write your answer as a fraction in simplest form. 19−(−29)=
The result of 19 minus a negative 29 is 48. Expressed as a fraction in simplest form, this would be 48/1.
To find the difference between 19 and negative 29, we can use the rule that subtracting a negative number is the same as adding its absolute value. So, 19 - (-29) is the same as 19 + 29, which equals 48.
To write this as a fraction in simplest form, we simply put 48 over 1, since any integer can be expressed as a fraction with a denominator of 1. We don't need to simplify any further, so our final answer is 48/1.
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Kimberly rolls two six-sided number cubes numbered 1 through 6 and adds up the two numbers construct a tree diagram to determine all the possible outcomes list the sum at the end of each branch of the tree
When Kimberly rolls two six-sided number cubes numbered 1 through 6, it creates 36 possible outcomes which is represent in the tree diagram below
What is a tree diagram?A tree diagram is a visual representation of outcomes. It consists of branches that represent the possible outcomes of each step.
When it comes to Kimberly rolling two six-sided number cubes, we can start by rolling the first cube, and then rolling the second cube.
For each roll of the first cube, there are six possible outcomes (1 to 6). For each outcome of the first cube, there are six possible outcomes for the second cube.
This results in a total of 6 x 6 = 36 possible outcomes.
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Find the value of x.
If necessary, round your answer to the nearest tenth.
O is the center of the circle.
The figure is not drawn to scale. Hint: Draw in the radius for both chords.
Remember radii are equal in the same circle.
FG I OP, RS 1 o.
FG = 25, RS = 28, OP = 19
R
P
19
S
The value of x is 26.5, found using the property of intersecting chords in a circle and the Pythagorean theorem.
How to find the value of x in a circle with intersecting chords?To find the value of x, we can use the property that states that if two chords intersect in a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.
In this case, we can draw radii from O to points P and S, and label their lengths as 19. Then, we can label the segments of chords FG and RS as follows:
Let a = FG and b = GP
Let c = RS and d = SP
Since OP is a radius of the circle, we know that a + b = 19. Similarly, since OS is a radius of the circle, we know that c + d = 19.
Using the property mentioned above, we can write:
a * b = c * d
Substituting the given values, we get:
25 * (19 - b) = 28 * (19 - d)
Expanding and simplifying, we get:
475 - 25b = 532 - 28d
Substituting a + b = 19 and c + d = 19, we get:
b = 19 - a and d = 19 - c
Substituting these values, we get:
25a - 25(19 - a) = 28c - 28(19 - c)
Simplifying, we get:
53a - 475 = 28c - 532
Rearranging, we get:
53a - 28c = -57
We also know that a + c = 25 + 28 = 53.
We can solve these two equations simultaneously to find the values of a and c:
a = 13.8
c = 39.2
Therefore, the length of the segment RS is 39.2, and the length of the segment RP19S is 58.2.
Using the Pythagorean theorem, we can find the length of the segment OP:
(OP)²= (RP19S)² - (19)²
(OP)² = (58.2)² - (19)²
(OP)² = 3136.24
OP = 56
Finally, we can find x using the fact that the chords FG and RS are parallel:
x = (1/2) * (FG + RS)
x = (1/2) * (25 + 28)
x = 26.5 (rounded to the nearest tenth)
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slove log2(x-6)+log2(x+6)=6
Answer: x = 10
Step-by-step explanation: To solve this equation, you can use the logarithmic property that states loga(b) + loga(c) = loga(bc). So, you can rewrite the left side of the equation as log2((x-6)(x+6)). Then, you can use the property that states loga(b) = c is equivalent to a^c = b to solve for x.
So, you have log2((x-6)(x+6)) = 6, which is equivalent to 2^6 = (x-6)(x+6). Simplifying the left side gives you 64, and expanding the right side gives you x^2 - 36 = 64. Solving for x gives you x = ±√100, which is x = ±10. However, since the original equation includes logarithms.
A chef uses the expression 0. 3n to represent the pounds of beef to use in a soup recipe for n people. The soup also has
7 fewer pounds of carrots than beef,
twice as many pounds of potatoes as beef, and
half as many pounds of onions as carrots.
How many pounds of ingredients will the chef use to make soup for 40 people?
Enter your answer in the box
The chef will use 43.5 pounds of ingredients to make soup for 40 people. Your answer is 43.5..
Step 1: Calculate the pounds of beef for 40 people using the expression 0.3n.
0.3 * 40 = 12 pounds of beef
Step 2: Calculate the pounds of carrots, which is 7 fewer than the pounds of beef.
12 - 7 = 5 pounds of carrots
Step 3: Calculate the pounds of potatoes, which is twice the pounds of beef.
2 * 12 = 24 pounds of potatoes
Step 4: Calculate the pounds of onions, which is half the pounds of carrots.
0.5 * 5 = 2.5 pounds of onions
Step 5: Add up the pounds of all ingredients to find the total pounds for 40 people.
12 (beef) + 5 (carrots) + 24 (potatoes) + 2.5 (onions) = 43.5 pounds
The chef will use 43.5 pounds of ingredients to make soup for 40 people. Your answer is 43.5.
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After some not so high practice dives by the circus owner, the circus performers decide to do a practice run of the show with the diver himself. but they decide to set it up so they will not have to worry about a moving cart. instead, the cart containing the tub of water is placed directly under the ferris wheel’s 11o’clock position. as usual, the platform passes the 3o’clock position at t=0
how many seconds will it take for the platform to reach the 11 o’clock position?
what is the diver’s height off the ground when he is at the 11 o’clock position?
radius = 50 ft
center of wheel is 65 feet off ground
turns counterclockwise at a constant speed, with a period of 40 seconds.
platform is at 3 o’clock position when it starts moving
The ferris wheel will take a total time of 10 seconds for the platform to reach the 11 o'clock position and the diver's height off the ground when he is at the 11 o'clock position is 50 feet.
To determine the time it takes for the platform to reach the 11 o'clock position and the diver's height off the ground, we will use the given information about the ferris wheel.
1. The ferris wheel has a radius of 50 ft and turns counterclockwise at a constant speed with a period of 40 seconds.
2. The center of the wheel is 65 ft off the ground.
3. The platform is at the 3 o'clock position when it starts moving (t=0).
The ferris wheel has a period of 40 seconds, which means it takes 40 seconds for it to make a full rotation. The distance between the 3 o'clock position and the 11 o'clock position is 90 degrees out of 360, which is one-fourth of the total distance around the circle.
Therefore, it will take 1/4 of the total time for the platform to reach the 11 o'clock position, which is 40/4 = 10 seconds.
To find the diver's height off the ground at the 11 o'clock position, we can use the sine function. Let's call the angle formed by the radius from the center of the ferris wheel to the diver and the radius from the center of the ferris wheel to the 3 o'clock position θ.
Since the platform starts at the 3 o'clock position and rotates counterclockwise, θ will increase as time passes. At the 11 o'clock position, θ will be 90 degrees.
We know that the radius of the ferris wheel is 50 feet and the center of the ferris wheel is 65 feet off the ground. Let's call the height of the diver off the ground h. Then we have:
sin θ = h / (50 ft)
h = (50 ft) * sin θ
At the 11 o'clock position, θ = 90 degrees, so we have:
h = (50 ft) * sin 90°
h = 50 ft
Therefore, the diver's height off the ground when he is at the 11 o'clock position is 50 feet.
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Krissa exercises daily by walking. she aims to walk at a steady rate of 3.4 miles per hour.
a. if t represents time in hours and d represents distance in miles, write an equation that models the relationship between these variables.
b. use your equation to calculate the distance krissa will jog in 5/8 of an hour (round to the hundredths).
c. use your equation to calculate how long it will take for krissa to walk 5.44 miles.
When Krissa exercises daily by walking and she aims to walk at a steady rate of 3.4 miles per hour:
a) The equation is d = 3.4t;
b) Krissa will walk 2.13 miles;
c) It will take Krissa 1.6 hours to walk 5.44 miles.
What is a) the equation for Krissa's walking speed (d = 3.4t), and b) how far will she walk in 5/8 of an hour (2.13 mi), and c) how long to walk 5.44 miles (1.6 hours)?When Krissa exercises daily by walking and she aims to walk at a steady rate of 3.4 miles per hour:
a. The equation that models the relationship between time and distance is:
d = 3.4t
where d is the distance Krissa walks in miles and t is the time she spends walking in hours.
b. To calculate the distance Krissa will walk in 5/8 of an hour, we can substitute t = 5/8 into the equation from part a:
d = 3.4(5/8) = 2.125 miles
Therefore, Krissa will walk 2.125 miles in 5/8 of an hour, rounded to the hundredths.
c. To calculate how long it will take Krissa to walk 5.44 miles, we can rearrange the equation from part a to solve for t:
t = d/3.4
Substituting d = 5.44 into this equation, we get:
t = 5.44/3.4 = 1.6 hours
Therefore, it will take Krissa 1.6 hours to walk 5.44 miles.
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Abby makes wants to make a gallon of punch. She uses 2 quarts of orange juice 1 cup of lemon juice and 2 1/2 pints of pineapple juice. How many cups of water should you add to make 1 gallon?
Abby wants to make 1 gallon (16 cups) of punch, she will need to add 16 - 14 = 2 cups of water to reach the desired amount.
To answer your question about how many cups of water Abby should add to make 1 gallon of punch, let's first convert all the given measurements to cups. One gallon is equivalent to 16 cups.
1. Orange juice: Abby uses 2 quarts of orange juice. Since there are 4 cups in a quart, she uses 2 x 4 = 8 cups of orange juice.
2. Lemon juice: Abby uses 1 cup of lemon juice.
3. Pineapple juice: Abby uses 2 1/2 pints of pineapple juice. There are 2 cups in a pint, so she uses (2 1/2) x 2 = 5 cups of pineapple juice.
Now, let's add up the cups of orange juice, lemon juice, and pineapple juice: 8 + 1 + 5 = 14 cups. Since Abby wants to make 1 gallon (16 cups) of punch, she will need to add 16 - 14 = 2 cups of water to reach the desired amount.
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The sum of the numerator and denominator of the fraction is 12. If the denominator is increased by 3, the fraction becomes 12. Find the fraction.
Let the fraction be x/y.
We know that x + y = 12, and that (x) / (y + 3) = 12.
Multiplying both sides of the second equation by (y + 3), we get:
x = 12(y + 3)
Substituting this into the first equation, we get:
12(y + 3) + y = 12
Expanding and simplifying, we get:
13y + 36 = 12
Subtracting 36 from both sides, we get:
13y = -24
Dividing both sides by 13, we get:
y = -24/13
Substituting this value of y into the equation x + y = 12, we get:
x - 24/13 = 12
Multiplying both sides by 13, we get:
13x - 24 = 156
Adding 24 to both sides, we get:
13x = 180
Dividing both sides by 13, we get:
x = 180/13
Therefore, the fraction is 180/13 divided by -24/13, which simplifies to -15/2.
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a non-degenerate triangle. what is the sum of the values of $k$ for which the area of the triangle is a minimum
The minimum value of k that satisfies this inequality is k = 9.
To find the value of k for which the area of the triangle is a minimum, we'll use the following terms: vertices, non-degenerate triangle, and area of a triangle. Here's the step-by-step explanation:
1. The vertices of the triangle are $(1, 7), (13, 16),$ and $(5, k)$.
2. A non-degenerate triangle means it has a positive area.
3. The area of a triangle can be calculated using the formula: Area = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Now, let's find the area of the triangle with the given vertices:
Area = (1/2) * |1(16 - k) + 13(k - 7) + 5(7 - 16)|
We want to minimize the area, so let's simplify the expression:
Area = (1/2) * |-9 + 13k - 104|
Since we want a non-degenerate triangle, the area must be greater than 0. Therefore, the expression inside the absolute value must be positive:
-9 + 13k - 104 > 0
13k > 113
k > 113/13
k > 8.69
Since k is an integer, the minimum value of k that satisfies this inequality is k = 9.
The sum of the values of k for which the area of the triangle is a minimum is just the single value we found, which is 9.
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Solve the inequalities 1/3(2x-1)≤1-2/5(2-3x)
The solution to the inequality is x ≥ -1.
We solve the inequality 1/3(2x-1)≤1-2/5(2-3x).
Let's go step by step:
Begin by distributing the fractions to the terms inside the parentheses:
(1/3 * 2x) - (1/3 * 1) ≤ 1 - (2/5 * 2) + (2/5 * 3x)
(2x/3) - (1/3) ≤ 1 - (4/5) + (6x/5)
Combine like terms on each side of the inequality:
(2x - 1)/3 ≤ (1 - 4/5) + 6x/5
(2x - 1)/3 ≤ (1/5) + 6x/5.
To eliminate the fractions, find a common denominator, which in this case is 15.
Multiply each term by 15:
15 * (2x - 1)/3 ≤ 15 * (1/5) + 15 * 6x/5
5(2x - 1) ≤ 3 + 18x
Distribute and simplify:
10x - 5 ≤ 3 + 18x
Move the variables to one side and constants to the other side:
10x - 18x ≤ 3 + 5
-8x ≤ 8
Divide both sides by -8 (remember to flip the inequality sign since we are dividing by a negative number):
x ≥ -1.
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A third candle, in the shape of a right circular cone, has a volume of 16 cubic inches and a radius of 1. 5 inches. What is the height, in inches, of the candle? Round your answer to the nearest tenth of an inch.
The height of the right circular cone ,candle is approximately 6.8 inches.
To find the height of the third candle, which is a right circular cone with a volume of 16 cubic inches and a radius of 1.5 inches, we will use the formula for the volume of a cone: V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.
1. Substitute the given values into the formula: 16 = (1/3)π(1.5)^2h
2. Simplify the equation: 16 = (1.5^2 * π * h) / 3
3. Solve for h:
a. Multiply both sides by 3: 48 = 1.5^2 * π * h
b. Divide by π: 48/π = 1.5^2 * h
c. Divide by 1.5^2: (48/π) / 1.5^2 = h
4. Calculate the height, and round to the nearest tenth: h ≈ 6.8 inches
The height of the candle is approximately 6.8 inches.
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I Need help with this (if you can’t see zoom in)
Answer: 45 (depends on what the rightmost angle is)
Step-by-step explanation:
All angles of a triangle add up to 180 degrees.
Two angle measures are provided, 100 and 35 (?)
Add up the two to get 135, and subtract from 180
180 - 135 = 45 degrees.
I might be seeing the rightmost angle measure wrong but I think it's 35, if it's not you can still apply the same strategy, just add the two given angles and subtract that from 180 to find x.
5x−4<10give your answer as an improper fraction in its simplest form.
The value of x as an improper fraction in its simplest form is 14/5.
To solve the inequality 5x - 4 < 10, we need to isolate x on one side of the inequality. First, we add 4 to both sides:
5x - 4 + 4 < 10 + 4
5x < 14
Then, we divide both sides by 5:
5x/5 < 14/5
x < 2.8
Therefore, the solution to the inequality is x < 2.8. However, the question asks for the answer as an improper fraction in its simplest form. To convert 2.8 to an improper fraction, we multiply both the numerator and denominator by 10 to get rid of the decimal:
2.8 * 10 / 1 * 10 = 28 / 10
To simplify the fraction, we divide both the numerator and denominator by their greatest common factor, which is 2:
28 / 10 = 14 / 5
Therefore, the answer is 14/5.
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You make street signs. This morning, you need to make a triangular sign and a circular sign. The material used to make the signs costs $17. 64 per square foot. Based on the designs, the base of the triangular street sign is 3 feet, and the height is 2. 6 feet. The circular street sign has a radius of 1. 5 feet. What is the total cost to make the two signs?
Answer: $193.42
Step-by-step explanation:
Based on the given designs, the cost to make the triangular and circular street signs would be $68.60 and $124.42 respectively, making the total cost of making both signs $193.02.
To calculate the cost of making the two street signs, we need to first find the area of each sign. The area of a triangle is given by the formula 1/2 x base x height. So, for the triangular street sign, the area would be 1/2 x 3 x 2.6 = 3.9 square feet.
The area of a circle is given by the formula π x radius². So, for the circular street sign, the area would be π x (1.5)² = 7.065 square feet.
Now that we have the areas of both signs, we can calculate the total cost of the material needed. The cost per square foot of material is $17.64, so we need to multiply this by the total area of the signs.
For the triangular sign, the cost would be 3.9 x $17.64 = $68.60.
For the circular sign, the cost would be 7.065 x $17.64 = $124.42.
Therefore, the total cost to make both signs would be $68.60 + $124.42 = $193.02.
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write an expression to represent; "The sum of a number b and 24"
Answer: ?
Answer:
b+24
Step-by-step explanation:
the sum of a number represented by variable b
-- b+
and 24
-- b+24
Answer: b + 24
Step-by-step explanation:
The sum of .. and ➜ addition between two values
a number b ➜ b (represented by a variable)
24 ➜ the number 24
The sum of a number b and 24 ➜ b + 24
CAN SOMEONE PLEASE HELP ME ILL GIVE BRAINLIST
Mai and Elena are shopping
for back-to-school clothes. They found a skirt that originally cost $30
on a 15% off sale rack. Today, the store is offering an additional 15% off. To find the new price of
the skirt, in dollars, Mai says they need to calculate 30. 0. 85 0. 85. Elena says they can just
multiply 30. 0. 70.
1. How much will the skirt cost using Mai's method?
2. How much will the skirt cost using Elena's method?
3. Explain why the expressions used by Mai and Elena give different prices for the skirt. Which
method is correct?
1. The skirt cost using Mai's method is $21.68.
2. The skirt cost using Elena's method is $21.
3. Mai's method is correct because she correctly calculates the discounts sequentially while Elena combines the discount.
We'll examine the methods suggested by Mai and Elena for finding the new price of the skirt and determine which one is correct.
1. Using Mai's method (30 x 0.85 x 0.85):
1: Calculate the first 15% off discount: 30 x 0.85 = 25.50
2: Calculate the additional 15% off discount: 25.50 x 0.85 = 21.675
So, the skirt will cost $21.68 using Mai's method (rounded to the nearest cent).
2. Using Elena's method (30 x 0.70):
Elena suggests taking 30% off the original price. To do this, we multiply the original price by 0.70:
30 x 0.70 = 21
So, the skirt will cost $21 using Elena's method.
3. Explanation of the difference in expressions and the correct method:
Mai's method is correct because she correctly calculates the discounts sequentially. The first 15% off is applied to the original price, and then the additional 15% off is applied to the reduced price. This results in a final price of $21.68.
Elena's method is incorrect because she combines the two discounts into a single 30% off, which does not accurately reflect the sequential discounts. By doing this, she finds a final price of $21, which is not correct.
In conclusion, Mai's method (30 x 0.85 x 0.85) is the correct way to calculate the new price of the skirt, resulting in a final cost of $21.68.
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Susan’s weekly earnings were proportional to the number of hours she worked. this table shows
the number of hours susan worked and the amount she earned. how much money did susan
earn per hour?
hours earnings ($)
5 $47.50
7 $66.50
9 $85.50
11 $104.50
Susan earns $9.50 per hour. This is found by dividing her earnings by the number of hours worked for each corresponding row in the table.
To find how much money Susan earned per hour, we need to divide the total earnings by the total number of hours worked. For finding the Total earnings we need to add the money earned in every hour,
Total earnings = $47.50 + $66.50 + $85.50 + $104.50 = $304
Total hours worked = 5 + 7 + 9 + 11 = 32
Money earned per hour = Total earnings / Total hours worked
= $304 / 32
= $9.50
Therefore, Susan earned money of $9.50 per hour.
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What's the solution?
The solution of the graphs of the equations is; )(6, 3 2/3)
What is a system of equation?A system of equation consists of two or more equations that share the same variables.
The solution of a system of equations obtained graphically can be obtained from the point of intersection of the lines of the graph of the equations
Taking the axis as the lowermost and leftmost white lines, we get;
The points on the function f are (0, 1), and (9, 5)
The slope is; (5 - 1)/(9 - 0) = 4/9
The y-intercept is; (0, 1)
The equation is; y = (4/9)·x + 1
The equation of the line g is; x = 6
Therefore, the point of intersection is; y = (4/9)×6 + 1 = 8/3 + 1 = 11/3 = 3 2/3
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Cory and Dalia like to buy fruit at the farmers’ market on Sundays. One Sunday, Cory bought 4 apples and 6 oranges and paid $5.10. Dalia bought 2 apples and 5 oranges and paid $3.65.
What is the cost of 2 oranges?
Write the answer as a decimal to 2 places
The cost of two oranges is 1.1 dollars.
How to find the cost of two oranges?One Sunday, Cory bought 4 apples and 6 oranges and paid $5.10. Dalia bought 2 apples and 5 oranges and paid $3.65.
Therefore, using equation,
let
x = cost of each apples
y = cost of each oranges
Hence,
4x + 6y = 5.10
2x + 5y = 3.65
Multiply equation(ii) by 2
4x + 6y = 5.10
4x + 10y = 7.3
4y = 2.2
y = 2.2 / 4
y = 0.55 dollars
Therefore,
cost of 2 oranges = 0.55(2) = 1.1 dollars
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The equation that represents the asymptote of the function, y = tan x is: C. x = π/4.
How to Determine the Equation that Represents the Asymptote of a Graph?Option A, x = -π, and Option D, x = (3π)/2, do not represent asymptotes of the graph of the function y = tan x.
Option B, x = 0, represents a vertical asymptote of the graph of y = tan x because tan x is undefined at x = π/2 + kπ, where k is an integer. Therefore, tan x is undefined at x = π/2, 3π/2, 5π/2, etc. and there is a vertical asymptote at x = 0.
Option C, x = π/4, represents a linear asymptote of the graph of y = tan x. As x approaches π/4 from either side, the tangent function approaches a straight line with slope 1 and x-intercept 0. Therefore, the equation of the asymptote is y = x - π/4.
Thus, the answer is C. x = π/4.
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After an antibiotic is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C(t) = 4te^{-39t}, where t is measured in ug/mg hours and C is measured in Use the closed interval methods to detremine the maximum concentration of the antibiotic between hours 1 and 7. Write a setence stating your result, round answer to two decimal places, and include units.
The maximum concentration of the antibiotic between hours 1 and 7 is 1.03 mg/ug, which occurs at t = 1/39 hours.
To find the maximum concentration of the antibiotic between hours 1 and 7, we need to find the maximum value of the function C(t) on the interval [1, 7]. We can do this by taking the derivative of C(t), setting it equal to zero, and solving for t.
C(t) = 4te^{-39t}
C'(t) = 4e^{-39t} - 156te^{-39t}
Setting C'(t) equal to zero, we get:
4e^{-39t} - 156te^{-39t} = 0
4e^{-39t}(1 - 39t) = 0
1 - 39t = 0
t = 1/39
We can now evaluate C(t) at t = 1/39 and the endpoints of the interval [1, 7] to determine the maximum concentration:
C(1) = 4e^{-39} ≈ 0.00011 mg/ug
C(7) = 28e^{-273} ≈ 0.000000003 mg/ug
C(1/39) = 1.0256 mg/ug
Therefore, the maximum concentration of the antibiotic between hours 1 and 7 is 1.03 mg/ug, which occurs at t = 1/39 hours.
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Define a useful quantity for our expectation about the amount of a time an
in-control process will remain ostensibly in-control, the average run length (ARL), to be the number of samples that will be observed, on average, before a point falls outside control limits. If p is the probability that any given point falls outside the control limits, then:
ARL = 1/p
(Estimating the true ARL may be useful for detecting an out-of-control process, but not
necessarily doing so quickly.) A process is Gaussian with mean 8 and standard deviation 2. The process is monitored by taking samples of size 4 at regular intervals. The process is declared to be out of control if a point plots outside the 3σ control limits on an X-chart. If the process mean shifts to 9, what is the average number of samples that will be drawn before the shift is detected on an X-chart?
Answer:
First, we need to calculate the control limits for the X-chart. Since the sample size is 4, the standard deviation of the sample mean is:
σ/√n = 2/√4 = 1
The 3σ control limits for the X-chart are:
Upper Control Limit (UCL) = 8 + 3(1) = 11
Lower Control Limit (LCL) = 8 - 3(1) = 5
Next, we need to find the probability that a point falls outside the control limits when the process mean shifts to 9. This can be calculated using the Gaussian distribution:
P(X < 5 or X > 11) = P(Z < (5-9)/2) + P(Z > (11-9)/2) = 0.00135 + 0.00135 = 0.0027
where Z is the standard normal distribution.
Therefore, the average run length (ARL) is:
ARL = 1/p = 1/0.0027 = 370.37
So on average, it will take 370.37 samples before the process mean shift is detected on an X-chart.
Find the exact length of the curve. 36y² = (x² – 4)³, 5 ≤ x ≤ 9, y ≥ 0 = 96.666
The exact length of the curve is 112/3(√3 + 1), or approximately 96.666.
To find the exact length of the curve, we can use the formula for arc length:
L = ∫a^b √(1 + [f'(x)]²) dx
where f(x) = (x² - 4)^(3/2)/6, 5 ≤ x ≤ 9.
First, we find f'(x):
f'(x) = 3x(x² - 4)^(1/2)/12 = x(x² - 4)^(1/2)/4
Then we substitute f'(x) into the formula for arc length:
L = ∫5^9 √(1 + [x(x² - 4)^(1/2)/4]²) dx
L = ∫5^9 √(1 + x²(x² - 4)/16) dx
L = ∫5^9 √(16 + 16x²(x² - 4)/16) dx
L = ∫5^9 √(16x² + x^4 - 4x²) dx
L = ∫5^9 √(x^4 + 12x²) dx
L = ∫5^9 x²√(x^2 + 12) dx
We can use the substitution u = x^2 + 12, which gives du/dx = 2x and dx = du/2x, to simplify the integral:
L = (1/2)∫37^93 √u du
L = (1/2) [(2/3)u^(3/2)]_37^93
L = (1/3)[(125 + 108√3) - (13 + 36√3)]
L = (1/3)(112√3 + 112)
L = 112/3(√3 + 1)
Therefore, the exact length of the curve is 112/3(√3 + 1), or approximately 96.666.
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Find the perimeter of a rectangle whose vertices are (3,2), (13,2), (13,9), and (3,9).
The perimeter of a rectangle is 34 units.
To find the perimeter of the rectangle with vertices (3,2), (13,2), (13,9), and (3,9), you need to determine the lengths of its sides.
The distance between (3,2) and (13,2) is the length of the base: 13 - 3 = 10 units.
The distance between (3,2) and (3,9) is the height: 9 - 2 = 7 units.
The perimeter of a rectangle is given by the formula P = 2*(length + width). In this case, P = 2*(10 + 7) = 2*17 = 34 units.
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Evaluate the integral dy (tan-'[y/8)) (64+y?) ( 1 + dy (tan-'(4/8)) (64+y?) =
Answer: ln|y/8| + C
Explanation:
First, we need to recognize that the derivative of arctan(x) is 1/(1+x^2). Therefore, the derivative of arctan(y/8) is 8/(64+y^2).
Now, using the substitution u = y/8, we can rewrite the integral as:
∫(1/u)(64+64u^2)(8/(64+64u^2))du
Simplifying, we get:
∫(1/u)du = ln|u| = ln|y/8|
Therefore, the final answer is:
ln|y/8| + C
where C is the constant of integration.
Write a function to describe the following scenario.
Jonathan is selling his old trading cards.
Each customer that buys gets the first
box they purchase for $10, and each
additional box for only $5.
y = [?]x + [?]
Answer:
y = 5x + 5
Step-by-step explanation:
If x is the number of boxes sold and y is the cost
The first box costs $10
Each additional box costs $5.
If the total number of boxes sold is x, then after selling the first box for $10, there will be x - 1 boxes left to be sold
The cost of x -1 boxes at $5 per box = 5(x - 1) = 5x - 5
Therefore for a total of x boxes sold the total cost, y in dollars is
y = 10 (for the first box) + 5x - 5 (for the remaining x - 1 boxes)
= 10 + 5x - 5
= 5 + 5x
which in standard form is written as
y = 5x + 5
We can verify our equation using specific numbers for x
For x = 1
y = 5 + 5(1) = 10 ; since only one box has been sold, the cost is fixed at $10
For x = 2 y = 5 + 5(2) = 5 + 10 = $15
This works out to since first box is sold at $10 and the second box at $5
Leave it to you to work out for other numbers