The derivative of y = x^(sin x) with respect to x is:
dy/dx = x^(sin x) * (cos x * ln(x) + sin x * (1/x)).
To find the derivative of y = x^(sin x) with respect to x using logarithmic differentiation, follow these steps:
1. Take the natural logarithm of both sides of the equation:
ln(y) = ln(x^(sin x))
2. Use the properties of logarithms to simplify:
ln(y) = sin x * ln(x)
3. Differentiate both sides with respect to x, using the chain rule and product rule:
(1/y) * dy/dx = cos x * ln(x) + sin x * (1/x)
4. Multiply both sides by y to solve for dy/dx:
dy/dx = y * (cos x * ln(x) + sin x * (1/x))
5. Substitute the original expression for y (y = x^(sin x)) back into the equation:
dy/dx = x^(sin x) * (cos x * ln(x) + sin x * (1/x))
So the derivative of y = x^(sin x) with respect to x is:
dy/dx = x^(sin x) * (cos x * ln(x) + sin x * (1/x)).
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A store sells cashews for $6. 00 per pound and peanuts for $3. 00 per pound. The manager decides to mix 20
pounds of peanuts with some cashews and sell the mixture for $4. 00 per pound. How many pounds of cashews
should be mixed with the peanuts so that the mixture will produce the same revenue as would selling the nuts
separately?
The amount of cashews needed to be mixed with the peanuts so that the mixture will produce the same revenue as selling the nuts separately is 10 pounds.
To solve this problem, we need to use the equation:
$3(20) + 6x = 4(20 + x)$
where x is the number of pounds of cashews needed.
First, we simplify the equation by multiplying:
$60 + 6x = 80 + 4x$
Then we isolate x by subtracting 4x from both sides and subtracting 60 from both sides:
$2x = 20$
Finally, we solve for x by dividing both sides by 2:
$x = 10$
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What is the quotient of the expression (5.04×1012)÷(6.3×109) written in scientific notation?
The quotient of the expression (5.04×10^12)÷(6.3×10^9) written in scientific notation is 8 × 10^2.
To divide two numbers written in scientific notation, we can divide their coefficients (the decimal parts) and subtract their exponents. So, we have:
(5.04 × 10^12) ÷ (6.3 × 10^9) = (5.04 ÷ 6.3) × 10^(12-9) = 0.8 × 10^3
Since 0.8 is less than 1, we can write this number in scientific notation by moving the decimal point one place to the right and subtracting 1 from the exponent:
0.8 × 10^3 = 8 × 10^2
Therefore, the quotient of the expression (5.04×10^12)÷(6.3×10^9) written in scientific notation is 8 × 10^2.
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The lateral area of a cone is 614cm squared. The radius is 16.2 cm. What is the slant height to the nearest tenth of a cm?
The slant height of the given cone is 16.36 cm.
What is the slant height?The length from the base to the peak along the "center" of a lateral face of an object (like a frustum or pyramid) is its slant height.
It is, in other words, the height of the triangle that a lateral face is a part of (Kern and Bland 1948, p.
So, calculate the slant height as follows:
614π = πr√h²+r²
614 = 16.2√h²+16.2²
614 = 262.44√h²
614/262.44 = h
2.33
Height = 2.33 cm
Then, slant height formula:
s=√(r² + h²)
s=√(16.2² + 2.33²)
s=√267.8689
s=16.36 cm
Therefore, the slant height of the given cone is 16.36 cm.
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Given the following triangle, If Sin F = 3/5 , then find the Cos D: A) 4/5 B) 4/3 C) 3/4 D) 3/5
If Sin F = 3/5 , then the value of Cos D is 4/5 (option a)
Let us consider the triangle in the given question. Since we are given that Sin F = 3/5, we know that the side opposite angle F is 3 and the hypotenuse is 5. Using Pythagoras theorem, we can find the length of the adjacent side as follows:
Opposite² + Adjacent² = Hypotenuse²
3² + Adjacent² = 5²
9 + Adjacent² = 25
Adjacent² = 16
Adjacent = 4
So we have found that the length of the adjacent side is 4. Now we can use the definition of cosine to find Cos D.
Cosine is defined as the ratio of the adjacent side to the hypotenuse. Therefore,
Cos D = Adjacent/Hypotenuse = 4/5
Hence, the answer is option A) 4/5.
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Mai  Drew does design shown below each rectangle in the design have the same area each rectangle is a what fraction of the area of the complete design 
Answer:
1/3
Step-by-step explanation:
There are 3 rectangles, so 1 rectangle is 1 out of 3 rectangles. So the fraction would be 1/3.
Each rectangle is 1/3 fraction of the complete design.
What are Fractions?Fractions are type of numbers which are written in the form p/q, which implies that p parts in a whole of q.
Here p, called the numerator and q, called the denominator, are real numbers.
Given is a design drawn by Mai.
The design consists of three rectangles.
Also, given that each of the rectangle has the same area.
This means that if we find one of the rectangle's area, multiply it by 3 and we will get the whole area.
Or in other words, if we find the whole area, then divide it by 3 to get each of the rectangle's area.
So the required fraction is 1/3.
Hence the fraction is 1/3.
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A group of students collected old newspapers for a recycling project. The data shows the mass, in kilograms, of old newspapers collected by each student.
23, 35, 87, 64, 101, 90, 45, 76, 105, 60, 55
98, 122, 49, 15, 57, 75, 120, 56, 88, 45, 100.
What percent of students collected between 49 kilograms and 98 kilograms of newspapers? Explain how you got to your solution
To find the percentage of students who collected between 49 and 98 kilograms of newspapers, we need to first count the number of students whose collection falls within this range. We can do this by sorting the data and counting the number of values that fall within this range.
Sorting the data, we get:
15, 23, 35, 45, 45, 49, 55, 56, 57, 60, 64, 75, 76, 87, 88, 90, 98, 100, 101, 105, 120, 122
We can see that there are 17 students whose collection falls within the range of 49 to 98 kilograms.
To find the percentage of students, we can divide the number of students whose collection falls within this range by the total number of students and then multiply by 100. The total number of students is the sum of the number of values in the two sets, which is 22 + 22 = 44.
Therefore, the percentage of students who collected between 49 and 98 kilograms of newspapers is:
17/44 * 100% ≈ 38.6%
So approximately 38.6% of the students collected between 49 and 98 kilograms of newspapers.
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The derivative of the function ds/dt of the function s = (tan² t - sec² t)⁵ is ...
The derivative of s with respect to t is:
ds/dt = 10(sec² t - tan t) * (tan² t - sec² t)⁴
How to find the derivative of the function?To find the derivative of s with respect to t, we will use the chain rule and the power rule of differentiation.
Let u = (tan² t - sec² t). Then, s = u⁵.
Using the chain rule, we have:
ds/dt = (du/dt) * (ds/du)
Now, we need to find du/dt and ds/du.
Using the chain rule again, we have:
du/dt = d/dt(tan² t - sec² t) = 2tan t * sec² t - 2sec t * tan t * sec t = 2sec² t * (tan t - sec t)
To find ds/du, we can simply apply the power rule:
ds/du = 5u⁴
Substituting these into the original equation for ds/dt, we get:
ds/dt = (2sec² t * (tan t - sec t)) * (5(tan² t - sec² t)⁴)
Therefore, the derivative of s with respect to t is:
ds/dt = 10(sec² t - tan t) * (tan² t - sec² t)⁴
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Twice the difference of a number and 3 is at most -24
Answer:
2(x - 3) < -24
x - 3 < -12
x < -9
The inequality that represents the given statement is 2(x-3) ≤ -24, where x is the unknown number.
The given statement can be translated into an inequality as "twice the difference of a number (x) and 3 is at most -24". Mathematically, this can be represented as 2(x-3) ≤ -24. Simplifying this inequality, we get 2x - 6 ≤ -24, or 2x ≤ -18, which gives x ≤ -9. Therefore, any number less than or equal to -9 satisfies the given statement.
For example, x = -10 satisfies 2(-10-3) = -26, which is less than or equal to -24. However, any number greater than -9 does not satisfy the given statement. For example, x = -8 gives 2(-8-3) = -22, which is greater than -24. Therefore, the solution set for the given inequality is x ≤ -9.
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Determine all of the values of that satisfy the condition cos =0 where 0 ≤ 0 <360°
All the value of of that satisfy the condition cos θ = - 1/2 ,
where 0 ≤ θ <360° are,
⇒ 120°, 240°
Given that;
Expression is,
cos θ = - 1/2
where 0 ≤ θ <360°
Since, cos θ = - 1/2
Hence, It belong in second and third quadrant.
So, cos θ = - 1/2
cos θ = 120°
cos θ = 240°
Thus, All the value of of that satisfy the condition cos θ = - 1/2 ,
where 0 ≤ θ <360° are,
⇒ 120°, 240°
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A business invests $25,000 in an account that earns 5.1% simple interest annually.
What is the value of the account after 4 years?
[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$25000\\ r=rate\to 5.1\%\to \frac{5.1}{100}\dotfill &0.051\\ t=years\dotfill &4 \end{cases} \\\\\\ A = 25000[1+(0.051)(4)] \implies A=25000(1.204)\implies A = 30100[/tex]
Jenny is planning an extended trip to russia.the table below shows a list of cities which she would like to visit during her stay, along with the amount of money she anticipates needing to spend on travel, lodging, and similar expenses in each city. all costs are given in russian rubles (rub). city cost (rub) izhevsk 4,721 novosibirsk 4,870 nizhny novgorod 6,920 moscow 5,485 chelyabinsk 5,217 saint petersburg 5,960 tolyatti 5,598 yaroslavl 4,901 even though jenny wants to go to each of these cities, her budget is only rub 33,000, so she recognizes that she must change her itinerary and choose not to visit some cities. which of the following pairs of cities will not put jenny back under budget if she drops them? a. moscow and chelyabinsk b. izhevsk and nizhny novgorod c. novosibirsk and saint petersburg d. yaroslavl and tolyatti
The pairs of cities that will not put Jenny back under budget if she drops them are:
a. Moscow and Chelyabinsk
b. Izhevsk and Nizhny Novgorod
To determine which pair of cities Jenny can drop without going over budget, we need to find the total cost of each pair and see if it exceeds her budget of rub 33,000.
a. Moscow and Chelyabinsk:
Total cost = 5,485 + 5,217 = 10,702 rubles
This pair of cities is over budget, so Jenny cannot drop any other cities if she wants to visit both Moscow and Chelyabinsk.
b. Izhevsk and Nizhny Novgorod:
Total cost = 4,721 + 6,920 = 11,641 rubles
This pair of cities is over budget, so Jenny cannot drop any other cities if she wants to visit both Izhevsk and Nizhny Novgorod.
c. Novosibirsk and Saint Petersburg:
Total cost = 4,870 + 5,960 = 10,830 rubles
This pair of cities is under budget, so Jenny can drop this pair without going over budget.
d. Yaroslavl and Tolyatti:
Total cost = 4,901 + 5,598 = 10,499 rubles
This pair of cities is under budget, so Jenny can drop this pair without going over budget.
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If f(9) = 9, f'(9) = 4, limit x→9√(f(x))-3/√(x)-3 =
A. 1
B. 1/4
C. 1/2
D. -1/2
The answer is A. 1. We can use L'Hopital's rule to evaluate the limit:
limit x→9√(f(x))-3/√(x)-3 = limit x→9 (f(x)-9)/(x-9) / (√(f(x))-3)/(√(x)-3)
Now, we know that f(9) = 9 and f'(9) = 4, so we can use the definition of the derivative to write:
f(x) - f(9) = f'(9)(x-9) + o(x-9)
where o(x-9) represents a term that goes to 0 faster than x-9 as x approaches 9. Plugging this into the numerator, we get:
f(x) - 9 = 4(x-9) + o(x-9)
Plugging this into the denominator, we get:
√(f(x)) - 3 = √(4(x-9) + o(x-9)) = 2√(x-9) + o(1)
√(x) - 3 = √(x-9) + o(1)
Therefore, the limit becomes:
limit x→9 (4(x-9) + o(x-9))/(√(x-9) + o(1)) / (2√(x-9) + o(1))/(√(x-9) + o(1))
Simplifying this expression, we get:
limit x→9 2(4(x-9) + o(x-9))/(√(x-9) + o(1))^2
limit x→9 8 + 2o(1)/(x-9)
As x approaches 9, the o(1) term goes to 0, so the limit becomes:
8 + 2*0/0 = 8
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Austin use a mold to make cone-shaped cupcakes. The diameter of the mold is 3 inches, and the height of the mold is 2 inches. If one cubic inch is about 0. 55 ounces, how many ounces will 10 cupcakes weigh? Use 3. 14 for pi. Round to the nearest tenth of an ounce
The 10 cupcakes will weigh approximately 25.9 ounces when one cubic inch is about 0. 55 ounces, the diameter of the mold is 3 inches and the height of the mold is 2 inches.
Given data:
Diameter of the mold = 3 inches
Radius = 3/2 = 1.5 inches
Height of the mold = 2 inches.
One cubic inch = 0.55 ounces
π = 3. 14
We need to find how many ounces will 10 cupcakes weigh. to find that we need to find the volume of a cone and the weight of one cupcake. we can find the volume of a cone given by the formula,
[tex]V = (1/3)πr^2h[/tex]
Where:
r = radius
h = height
By Substituting the r and h values into the formula we get:
[tex]V = (1/3)πr^2h[/tex]
[tex]= (1/3) π ((1.5)^2) × (2)[/tex]
[tex]= (1/3) π×(2.25)×(2)[/tex]
= 1.5π
When one cubic inch of the cone is about 0.55 ounces, the weight of one cupcake is approximately
= 1.5π × 0.55
= 0.825π ounces.
The weight of 10 cupcakes is determined by multiplying by the weight of one cupcake, it is given as:
= 10 × 0.825π
= 8.25π ounces
= 8.25 × 3.14
= 25.9 ounces.
Therefore, the 10 cupcakes will weigh approximately 25.9 ounces.
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What is a good percentage (in decimal form) to multiply your earning to estimate your paycheck?
To estimate your paycheck, a good percentage to multiply your earning by would be 0.75 or 75%. When calculating your paycheck, it's important to account for taxes, deductions, and other withholdings that may be taken out of your gross pay.
This accounts for taxes, deductions, and other withholdings that are typically taken out of your paycheck before you receive your net pay. For example, if you earn $1,000 per pay period, multiplying by 0.75 would give you an estimated net pay of $750. However, keep in mind that this is just an estimate and your actual net pay may vary depending on your specific tax situation and other factors.
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PLEASE HELP WILL MARK BRANLIEST!!!
The customer can choose from 21 different gifts if they want to order only 1 item, 138 different gifts if they want to order 2 items of different kinds, and 280 different gifts if they want to order 3 items of different kinds.
To find the different number of gifts, we use the fundamental counting principle, which states that the total number of ways to choose a sequence of items is equal to the product of the number of choices available for each item.
⇒ If a customer wants to order only 1 item, they can choose from:
10 varieties of "greeting-cards"
4 types of "fruit-baskets"
7 flower arrangements
So, total number of different gifts that can be created is: 10 + 4 + 7 = 21.
⇒ If a customer wants to order 2 items of different kinds, they can choose from:
10 varieties of greeting-cards and 4 types of fruit baskets :
(10 × 4 = 40 choices)
10 varieties of greeting cards and 7 flower arrangements :
(10 × 7 = 70 choices)
4 types of fruit-baskets and 7 flower arrangements :
(4 × 7 = 28 choices)
So, number of different gifts that can be created is: 40 + 70 + 28 = 138.
⇒ If a customer wants to order 3 items of different kinds, they can choose from:
10 varieties of greeting cards, 4 types of fruit baskets, and 7 flower arrangements (10 × 4 × 7 = 280 choices),
So, the total number of different gifts that can be created is : 280.
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Use the image to determine the type of transformation shown.
Image of polygon ABCD and a second polygon A prime B prime C prime D prime below.
The type of transformation shown is a vertical transformation
Determining the type of transformation shown.In mathematics, a vertical translation refers to a transformation of a function or graph that involves moving it up or down along the y-axis without changing its vertical transformation.
From the attached figure, we can see that the polygons are moved up or down to map them to one another
This means that the transformation is a vertical transformation
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Complete question
Use the image to determine the type of transformation shown.
Image of polygon ABCD and a second polygon A prime B prime C prime D prime below.
Vertical translation
Reflection across the x-axis
180° counterclockwise rotation
Horizontal translation
A man spent 500 dollars on a shopping trip to Erewhon. If the milk costed 132 dollars and the chicken costed 220, how much did the fish cost?
Answer:148
Step-by-step explanation:
Assuming he only bought milk, fish and chicken.
220+132=352
500-352=148
Are the events mutually exclusive?
event a: rational numbers
event b: irrational numbers
no - overlapping
yes - mutually exclusive
Answer:
Yes they are mutually exclusive
Step-by-step explanation:
You cannot both be a rational number and an irrational number
The first term of a pattern is 509. The pattern follows the "subtract 7" rule. Which number is a term in the pattern?
A:516
B:500
C:495
D:464
Answer:
C
Step-by-step explanation:
first fine the nth term
a+(n-1)d
509+7n+7
516-7n
then equate the ans to the nth term
495=516-7n
7n=516-495
7n= -21
n= -3
17. Determine whether the two figures are similar. If so, give the similarity ratio of *
the smaller figure to the larger figure. The figures are not drawn to scale.
6 ft
8 ft
6 ft
Yes; 3:5
Yes; 2:3
OYes; 2:5
10 ft
12 ft
10 ft
No they are not similar
The answer is "No they are not similar". The correct option is D.
To determine if the two cuboids are similar, we need to check if their corresponding sides are proportional.
If we compare the corresponding sides of the first cuboid and the second cuboid, we get:
6/10 = 0.6
8/12 = 0.666...
6/10 = 0.6
Since these ratios are not equal, the corresponding sides are not proportional, and therefore the two cuboids are not similar.
Therefore, the answer is "No they are not similar".
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Calculate the derivatives of all orders: f'(x), F"(x), F"(x), f(4)(x), ..., f(n)(x), ... f(x) = (-2x + 1)3 f'(x) f''(x) = f''(x) = f(4)(x) = f(n) (x) for all n 25
The first derivative of f(x) is f'(x) = -12(-2x + 1)2. The second derivative is f''(x) =48(-2x + 1), and all higher derivatives have the form f^(n)(x) = (-1)n * 6 * n! * (-2x + 1)^(3-n).
To calculate the derivatives of all orders for f(x) = (-2x + 1)3, we first need to find the first derivative:
f(x) = (-2x + 1)³
f'(x) = 3(-2x + 1)²(-2)
f'(x) = 3(-2x + 1)²(-2)
f'(x) = -12(-2x + 1)2
Next, we find the second derivative:
f''(x) = d/dx(-12(-2x + 1)²)
f''(x) = 2(-2)(-12)(-2x + 1)
f''(x) = -12[2(-2x + 1)(-2)]
f''(x)= 48(-2x + 1)
We can continue this process to find the third and fourth derivatives:
f'''(x) = d/dx(96(-2x + 1))
f'''(x) = -384
f''''(x) = d/dx(-384)
f''''(x) = 0
Notice that the fourth derivative is 0, meaning that all higher derivatives will also be 0.
This is because the original function is a polynomial of degree 3, so its fourth derivative will be the derivative of a constant, which is 0.
Therefore, we can conclude that:
f(4)(x) = 0
f(n)(x) = 0 for all n ≥ 4.
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Boris's backyard has cement and grass. Find the area of the part with cement.
(Sides meet at right angles. )
4 m
1 m
2 m
grass
5 m
3 m
cement
5 m
The constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.
To find the area of the part with cement, we need to find the area of the entire rectangle and then subtract the area of the part with grass.
The area of the rectangle is: length x width = 5 m x 4 m = 20 m²
The area of the part with grass is: length x width = 3 m x 2 m = 6 m²
So the area of the part with cement is:
20 m² - 6 m² = 14 m²
Therefore, the area of the part with cement is 14 square meters.
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As the new owner of a supermarket, you have inherited a large inventory of unsold imported Limburger cheese, and you would like to set the price to that your revenue from selling it is as large as possible. Previous sales figures of the cheese are shown in the following table. Use the sales figures for the prices S3 and $5 per pound to construct a demand function of the form q = Ae^-bp, where A and b are constants you must determine. (Round A and b to two significant digits.) q = Use your demand function to find the price elasticity of demand at each of the prices listed. (Round your answers to two decimal places.) P = $3, E = P = $4, E = P = $5, E = At what price should you sell the cheese in order to maximize monthly revenue (Round your answer to the nearest cent.) $ If your total Inventory of cheese amounts to only 200 pounds, and It win spoil one month from now, how should you price it in order to receive the greatest revenue? (Round your answer to the nearest cent.) $ Is this the same answer you got In part (c)? If not, give a brief explanation. It is a higher price than in part (c) because at a lower price you cannot satisfy the demand. It is the same price. It is a lower price than in part (c) because at a higher price the demand is not high enough.
a) The demand function is 134.33e^-0.693p
b) At P = $3, we have elasticity is 0.83, at P = $4, we have elasticity is 1.05, at P = $5, we have elasticity is 1.34.
c) We should sell the cheese at a price of $3.84 per pound to maximize monthly revenue.
d) We should sell the cheese at a price of $4.22 per pound to generate the highest revenue within the timeframe of one month.
a) To construct a demand function of the form q = Ae^-bp, we can use the sales figures for the prices $3 and $5 per pound. First, we calculate the values of A and b:
A = q/p = 403/3 ≈ 134.33
b = ln(q/Ap) / p = ln(403/134.33) / (3-5) ≈ 0.693
Using these values, the demand function becomes:
q = 134.33e^-0.693p
b) To find the price elasticity of demand at each of the prices listed, we can use the formula:
E = (dq/dp) * (p/q)
At P = $3, we have:
E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 3) / 403 ≈ 0.83
At P = $4, we have:
E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 4) / 284 ≈ 1.05
At P = $5, we have:
E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 5) / 225 ≈ 1.34
c) To find the price that will maximize monthly revenue, we can use the formula:
p = (1/b) * ln(A/b)
Plugging in the values of A and b that we calculated earlier, we get:
p = (1/0.693) * ln(134.33/0.693) ≈ $3.84
d) If we only have 200 pounds of cheese and it will spoil in one month, we need to sell it at a price that will generate the highest revenue within that timeframe. To do this, we can use the formula:
R = pq
where R is the revenue, p is the price per pound, and q is the quantity sold. We can express q in terms of p using our demand function:
q = 134.33e^-0.693p
Substituting this into the revenue equation, we get:
R = p * 134.33e^-0.693p
To find the price that will maximize revenue, we can take the derivative of R with respect to p and set it equal to zero:
dR/dp = 134.33e^-0.693p - 93.13pe^-0.693p = 0
Solving this equation numerically, we get:
p ≈ $4.22
This price is different from the price calculated in part (c) because we have a limited quantity of cheese that will spoil, so we need to balance the price and quantity sold to maximize revenue within the given timeframe.
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Can Someone Help me with this It is not easy
[ 40 POINTS]Ф
Answer:
4^9262144Step-by-step explanation:
You get 4 pennies for a first job, 16 pennies for the second job, 64 pennies for the 3rd job, and you want to know how many pennies you get for the 9th job, if each job quadruples the pay.
Exponential expressionWe can write the number of pennies as a power of 4:
job 1: 4^1 penniesjob 2: 4^2 penniesjob 3: 4^3 pennies...job 9: 4^9 penniesYou will get 4^9 pennies for the 9th job.
That is 262144 pennies.
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12. A normal distribution has a mean of 34 and a standard deviation of 7. Find the range of
values that represent the middle 95% of the data.
F. 27
G. 20 X 48
H. 13
J. 6
The range of values that represent the middle 95% of the data is from 20.18 to 47.82 or (20.18, 47.82).
What is Hypothesis test?A measurable speculation test is a strategy for factual deduction used to conclude whether the information within reach adequately support a specific speculation. We can make probabilistic statements about the parameters of the population thanks to hypothesis testing.
According to question:The middle 95% of a normal distribution is located within 1.96 standard deviations from the mean in both directions.
Therefore, the lower limit is:
34 - 1.96(7) = 20.18
And the upper limit is:
34 + 1.96(7) = 47.82
So the range of values that represent the middle 95% of the data is from 20.18 to 47.82 or (20.18, 47.82).
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Can I have help with this question, Please?
"Write an equation for the ellipse."
Center: (1, -3)
The equation of the ellipse with center (1, -3), and a semi major axis of 4, and a semi minor axis of 3 is; (x - 1)²/4² + (y + 3)²/3²
What is the standard form of the equation of an ellipse?The standard form equation of an ellipse can be presented as follows;
(x - h)²/a² + (y - k)²/b² = 1
Where;
(h, k) = The coordinates of the center of the ellipse.
The length of the semi major axis = a
Length of the semi minor axis = b
The center of the specified ellipse is; (h, k) = (1, -3)
The semi major axis = 4
The length of the semi minor axis = 3
The equation of the ellipse is therefore;
(x - 1)²/4² + (y - (-3))²/3² = 1
(x - 1)²/4² + (y + 3)²/3² = 1
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In ΔPQR, the measure of ∠R=90°, the measure of ∠P=26°, and PQ = 8. 5 feet. Find the length of QR to the nearest tenth of a foot
In ΔPQR, the length of QR is approximately 4.1 feet to the nearest tenth of a foot.
In ΔPQR, given that ∠R=90°, ∠P=26°, and PQ=8.5 feet, you want to find the length of QR to the nearest tenth of a foot.
Step 1: Since ∠R is a right angle (90°), we can use trigonometric ratios to find QR. First, let's find ∠Q. We know that the sum of angles in a triangle is 180°, so ∠Q = 180° - (∠P + ∠R) = 180° - (26° + 90°) = 64°.
Step 2: Now that we have all the angles, we can use the sine formula to find QR. We'll use the sine of ∠P (26°) and the given side PQ (8.5 feet) as our reference. The sine formula is:
QR = (PQ * sin(∠P)) / sin(∠Q)
Step 3: Plug in the known values:
QR = (8.5 * sin(26°)) / sin(64°)
Step 4: Calculate the sine values and the division:
QR = (8.5 * 0.4384) / 0.8988 ≈ 4.1326
Step 5: Round the answer to the nearest tenth of a foot:
QR ≈ 4.1 feet
In ΔPQR, the length of QR is approximately 4.1 feet to the nearest tenth of a foot.
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1. Find the first derivative of x2/3 + y2/3 = k1 2. Find the first derivative of x cos(k1 x + k2 y) = y sen x.
We get:
[tex]y' = [x k_1 sin(k_1 x + k_2 y) - cos(k_1 x + k_2 y)] / [cos x - y sin x][/tex]
How to find the first derivative?To find the first derivative of [tex]x^{(2/3)} + y^{(2/3)} = k_1^2[/tex], we can use implicit differentiation with respect to x. Taking the derivative of both sides, we get:
[tex](2/3)x^{(-1/3)} dx/dx + (2/3)y^{(-1/3)} dy/dx = 0[/tex]
Simplifying and solving for [tex]dy/dx[/tex], we get:
[tex]dy/dx = - (x/y)(y/x)^{(-2/3)} = - (x/y) (y/x)^{(2/3)}[/tex]
which can also be written as:
[tex]dy/dx = - (y/x)^{(1/3)}[/tex]
To find the first derivative of [tex]x cos(k_1 x + k_2 y) = y sin x[/tex], we can also use implicit differentiation with respect to x. Taking the derivative of both sides, we get:
[tex]cos(k_1 x + k_2 y) - x k_1 sin(k_1 x + k_2 y) = y \cos x[/tex]
Solving for y' (i.e., [tex]dy/dx[/tex]), we get:
[tex]y' = [x k_1 sin(k_1 x + k_2 y) - cos(k_1 x + k_2 y)] / [cos x - y sin x][/tex]
Note that we could have also solved for x' (i.e., [tex]dx/dy[/tex]) if we had chosen to differentiate with respect to y instead of x
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Given the measure of an acute angle in a right triangle, we can tell the ratios of the lengths of the triangle's sides relative to that acute angle.
Here are the approximate ratios for angle measures
55
°
55°55, degree,
65
°
65°65, degree, and
75
°
75°75, degree.
Angle
55
°
55°55, degree
65
°
65°65, degree
75
°
75°75, degree
adjacent leg length
hypotenuse length
hypotenuse length
adjacent leg length
start fraction, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction
0.57
0.570, point, 57
0.42
0.420, point, 42
0.26
0.260, point, 26
opposite leg length
hypotenuse length
hypotenuse length
opposite leg length
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction
0.82
0.820, point, 82
0.91
0.910, point, 91
0.97
0.970, point, 97
opposite leg length
adjacent leg length
adjacent leg length
opposite leg length
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, end fraction
1.43
1.431, point, 43
2.14
2.142, point, 14
3.73
3.733, point, 73
Use the table to approximate
m
∠
L
m∠Lm, angle, L in the triangle below.
3.2
3.2
11.9
11.9
L
L
K
K
J
J
Choose 1 answer:
The angle measure of L in the triangle is approximately 75°.
Based on the given table, we can see that the ratio of the opposite leg length to the adjacent leg length for an angle measure of 75° is approximately 3.73. Looking at the triangle in the question, we can see that the side opposite to angle L is the hypotenuse and the adjacent leg is LK.
Therefore, the ratio of the opposite leg length to the adjacent leg length for angle L is equal to the ratio of the hypotenuse length to the length of segment LK.
From the figure, we can see that the length of segment LK is approximately 3.2 units. Therefore, the length of the hypotenuse is approximately 3.73 times the length of segment LK, or:
hypotenuse length ≈ 3.73 × 3.2 ≈ 11.9
Therefore, the angle measure of L is approximately 75°.
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Write the repeating decimal as a geometric series. 0,216
the repeating decimal 0.216 can be written as the geometric series: 0.216 = 216/990.
To write the repeating decimal 0.216 as a geometric series, we first need to express it in the form of a sum of a geometric series.
The decimal 0.216 repeats every three digits, so we can break it down as follows:
0.216 = 0.2 + 0.01 + 0.006 + 0.0002 + 0.00001 + 0.000006 + ...
Now, we can write this as a sum of a geometric series with the first term (a) and the common ratio (r):
a = 0.2
r = 0.01 (because each term is 1/100 of the previous term)
Thus, the geometric series for the repeating decimal 0.216 is:
0.216 = 0.2 + 0.2(0.01) + 0.2(0.01)^2 + 0.2(0.01)^3 + ...
The formula for the sum of an infinite geometric series is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
Using the values for a and r, we can find the sum of the series:
S = 0.2 / (1 - 0.01) = 0.2 / 0.99 = 216/990.
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