The maximal area of an isosceles triangle-shaped field by the river is achieved when the triangle is also a right triangle. The area can be maximized by using the derivative of the area expression, considering the sides of the triangle and the fixed perimeter.
Explanation:The question is asking to find the maximal area of an isosceles triangle-shaped field with two equal sides and the third side being the river.
To maximize the area of an isosceles triangle, the height of the triangle should be as large as possible, which occurs when the triangle is also a right triangle. The side opposite the right angle, which lies along the river, will be the base of the triangle.
By applying the Pythagorean theorem, if we have a fixed perimeter and the two equal sides are 'a' and 'a', and the base 'b' is along the river, we can express the area 'A' as A = (b/2) × √(a² - (b/2)²).
The area is maximized when the derivative of this area expression with respect to 'b' equals zero.
What is the product of 5.1 × 10–8 and 0.07?
What is the equation of a line with a slope of 3 and a point
(3, 1) on the line?
Express the equation in the form of
y=mx+b where m is the slope and b is the y-intercept.
Divide. 2\3÷4\5 2\8 8\15 5\6 8\8
Which function in vertex form is equivalent to f(x) = x2 + 8 – 16x?
A. f(x) = (x – 8)2 – 56
B. f(x) = (x – 4)2 + 0
C. f(x) = (x + 8)2 – 72
D.f(x) = (x + 4)2 – 32
Answer: The correct option is A.
Explanation:
The given function is,
[tex]f(x)=x^2+8-16x[/tex]
Rewrite the above function.
[tex]f(x)=(x^2-16x)+8[/tex]
To make the perfect square we add and subtract the square of [tex]\frac{b}{2a}[/tex], where b is coefficient of x and a is the coefficient of [tex]x^2[/tex].
Since a=1 and b = -16, So we will add and subtract he square of -8.
[tex]f(x)=(x^2-16x+(-8)^2)+8-(-8)^2[/tex]
[tex]f(x)=(x^2-16x+(8)^2)+8-64[/tex]
Using [tex](a-b)^2=a^2-2ab+b^2[/tex]
[tex]f(x)=(x-8)^2)-56[/tex]
Therefore, the correct option is A.
Aiko started jump rope for an amazing 20 minutes, she stopped at 8;05.When did she start jumping
Aiko started jumping rope at 7:45, which is calculated by subtracting the 20-minute jump rope activity duration from the time she stopped at 8:05.
Explanation:To solve this problem, we need to subtract the duration of Aiko's jump rope activity from the time she stopped. We know that Aiko stopped jumping rope at 8:05 and the duration of her jump rope activity was 20 minutes.
Since Aiko's activity lasted 20 minutes and she ended at 8:05, we need to subtract 20 minutes from 8:05 to find out when she started. Thus, Aiko started jumping rope at 7:45.
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You are saving money to buy a new bicycle that costs $155.75. You have $30 and plan to save $5 each week. Your aunt decides to give you an additional $10 each week. a. How many weeks will you have to save until you have enough money to buy the bicycle? You need to save for how many weeks? b. How many more weeks would you have to save to buy a new bicycle that costs $203.89? You would have to save for how many more weeks?
Answer:
a). 9 weeks
b). 4 more weeks
Step-by-step explanation:
a). The cost of new bicycle = $155.75
Amount I have with me = $30
Therefore remaining amount = 155.75 - 30
= $ 125.75
Amount I will save in one week = $ 5
Amount my aunt will give me one week = $ 10
Therefore total amount i will save in one week = 5 + 10
= $ 15 in one week
Therefore number of weeks required to collect the remaining amount of $125.75 is = 125.75 / 15
= 8.38 weeks
= 9 weeks
Thus 9 weeks are required to save money to buy a bicycle that cost $155.75
b). New cost the bicycle = $ 203.89
From above at the end of 9 weeks I will have = $ 125.75
Amount I have with me before = $ 30
Therefore by the end of 9 weeks I have total amount = 125.75 + 30
= $ 155.75
Therefore amount less = 203.89 - 155.75
= $ 48.14
Number of weeks require to collect the remaining amount of $ 48.14 by saving $ 15 in one week is = 48.14 / 15
= 3.20 week
= 4 weeks
Thus, 4 more weeks is required to save money to buy a new bicycle that cost $ 203.89
It will take approximately 9 weeks to save enough money to buy the bicycle. It would take approximately 12 more weeks to save enough money for a bicycle that costs $203.89.
Explanation:To determine how many weeks it will take to save enough money to buy a bicycle, we can set up an equation:
$30 + ($5 + $10) imes x = $155.75
Where x represents the number of weeks. Now, we can solve for x:
$30 + $15x = $155.75
$15x = $125.75
x = $125.75 / $15 = 8.38
Since we can't have a fraction of a week, we can round up to the nearest whole number. So, it will take approximately 9 weeks to save enough money.
To determine how many more weeks you would have to save to buy a bicycle that costs $203.89, we can again set up an equation:
$30 + ($5 + $10) imes x = $203.89
Now, we can solve for x:
$30 + $15x = $203.89
$15x = $173.89
x = $173.89 / $15 = 11.59
Rounding up to the nearest whole number, it would take approximately 12 more weeks to save enough money.
How do I factor 27x^3 -125? The answer is not
(3x+5)(9x^2 -15x+25) or
(3x-5)(9x^2 +10x +25)
Answer: the first one
Step-by-step explanation:
B) suppose the average fisherman can catch 200,000 pounds of bluefin tuna every year (100 tons). what is the value of a bluefin tuna fishing license?
The value of a bluefin tuna fishing license depends on the price of bluefin tuna and the annual catch limit. The value can be significant, given the potential income of $650,000 per year from tuna fishing. However, it represents a balance between profit and sustainable fishing practices.
Explanation:The value of a bluefin tuna fishing license mostly depends on the price of bluefin tuna. Given that the price is $3.25 per pound and the average fisherman can catch 200,000 pounds (100 tons) per year, a fisherman could potentially earn $650,000 from tuna fishing annually. This suggests that the value of a fishing license may be quite high. However, the exact cost of the license is determined by several factors, including regulations, environmental impacts (often referred to as a 'tragedy of the commons' scenario), and maintaining a sustainable fishing industry.
Fisheries supposedly operate by established catch limits to prevent decimating the bluefin tuna population, meaning they are restricted in how much fish they can catch. Thus, the value of a fishing license also includes the advantage of sustainable fishing practices, ensuring the bluefin tuna's availability for future generations.
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how does $4.30 relate 4.3
5210÷17 show your work
what is the principal or(postive)square root of a number N
Find the jacobian of the transformation. x = u2 + uv, y = uv2
To find the Jacobian of the given transformation, we need to compute the partial derivatives of x and y with respect to u and v. The Jacobian matrix is a matrix that represents these partial derivatives.
Explanation:To find the Jacobian of the transformation x = u^2 + uv and y = uv^2, we need to compute the partial derivatives of x and y with respect to u and v. Let's start with the partial derivative of x:
∂x/∂u = 2u + v
Now let's find the partial derivative of x with respect to v:
∂x/∂v = u
Using the same process, we can find the partial derivatives of y:
∂y/∂u = v^2
∂y/∂v = 2uv
Putting it all together, the Jacobian matrix is:
J(u, v) = [∂x/∂u, ∂x/∂v; ∂y/∂u, ∂y/∂v]
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The Jacobian of the transformation is [tex]\( uv(4u + v) \)[/tex].
The Jacobian of the transformation given by [tex]\( x = u^2 + uv \) and \( y = uv^2 \)[/tex] is determined by finding the determinant of the matrix of partial derivatives. The matrix of partial derivatives, known as the Jacobian matrix, is given by: [tex]J = \begin{bmatrix} \frac{\partial x}{\partial u} \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} \frac{\partial y}{\partial v} \end{bmatrix}[/tex]
We calculate each of the partial derivatives:
[tex]\frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(u^2 + uv) = 2u + v[/tex]
[tex]\frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(u^2 + uv) = u[/tex]
[tex]\frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(uv^2) = v^2[/tex]
[tex]\frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(uv^2) = 2uv[/tex]
Now we can construct the Jacobian matrix: [tex]J = \begin{bmatrix} 2u + v u \\ v^2 2uv \end{bmatrix}[/tex]
The determinant of this matrix gives us the Jacobian of the transformation:
[tex]\text{Jacobian} = \text{det}(J) = (2u + v)(2uv) - (u)(v^2)[/tex]
[tex]\text{Jacobian} = 4u^2v + 2uv^2 - uv^2[/tex]
[tex]\text{Jacobian} = 4u^2v + uv^2[/tex]
[tex]\text{Jacobian} = uv(4u + v)[/tex]
Therefore, the Jacobian of the transformation is [tex]\( uv(4u + v) \).[/tex]
15 3/4% is equal to which decimal?
A- 0.1575
B- 157.25
C-15.25
D- 15.34
The function d(x) = 1375 − 110x represents the distance (in miles) a high-speed
train is from its destination after x hours. (Section 3.3)
a. How far is the train from its destination after 8 hours?
b. How long does the train travel before reaching its destination?
We are given the function:
d (x) = 1375 – 110 x
A. To solve for the distance between the train and its destination after 8 hours, we simply have to plug in, x = 8 into the equation. That is:
d (x) = 1375 – 110 * 8
d (x) = 495 miles
Therefore the train is now only 495 miles away from the destination.
B. To solve for the total time needed to reach its destination, we must set d = 0 and find for the value of x:
0 = 1375 – 110 x
110 x = 1375
x = 12.5 hours
Therefore it takes 12.5 hours for the train to reach its destination.
Solve for x.
6x−24=x−2
What is the answer to this question ?
Chuanxi planned a rectangular sidewalk with a length of 21 ft. He made a scale drawing using a scale factor of 1 in. = 7 ft. He decided to change the length of the actual sidewalk to 27 ft. If the scale drawing still has a length of 3 in., what does 1 in. represent in the new scale?
The unit is converted in new scale as 1 inch is equal to 9 feet.
What is unit conversion?Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.
For the given situation,
A rectangular sidewalk has a length = 21 feet
Scale factor, 1 inch = 7 feet.
So, 21 feet = [tex]\frac{21}{7}[/tex]
⇒ [tex]3[/tex] inches
The length of the actual side walk changes to 27 feet.
The length of actual side walk in inches = 3 inches.
Now, 1 inch = [tex]\frac{27}{3}[/tex]
⇒ [tex]1 inch = 9 feet[/tex]
Hence we can conclude that the unit is converted in new scale as 1 inch is equal to 9 feet.
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Noah visited 40 people in 4 hours. How long will it take him to visit 120 people?
The special mix contains 5/6 pound of dried chicken, 2/3 pound of dried bison, and 1/2 pound of dried vegetables.
How many pounds of special mix does Max get? Explain how you found your answer.
total pounds
to add, find a common deonmenator (bottom numbers)
the denomenantors are 6,3,2
the common denominator is 6 because 6*1=6, 3*2=6, 2*3=6
multiply each by 1 or a/a where a=a to make the common denomenators so we can add
remember that [tex]\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}[/tex]
5/6 times 1/1=5/6
2/3 times 2/2=4/6
1/2 times 3/3=3/6
so
[tex]\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=[/tex]
[tex]\frac{5}[6}+\frac{4}[6}+\frac{3}{6}=[/tex]
[tex]\frac{5+4+3}{6}=[/tex]
[tex]\frac{12}{6}=[/tex]
[tex]2[/tex]
2 pounds total of special mix
I found my answer using maths
neal buys a board games. he pays for the board game and pays $1.54 in sales tax. the sales tax rate is 5.5%. what is the original price of the board game, before tax?
how do you work out 21120÷5280 in long division?
At his comic book store, Korey’s Comics, Korey sells approximately $3,250 in comic books each month. But as a comic book dealer, Korey only pays $1,285 for these comic books. Korey’s monthly operating expenses, including labor, are $875. Calculate Korey’s monthly net income.
a. $1,090
b. $1,965
c. $2,375
d. $2,840
Answer:
(A) $1,090
Step-by-step explanation:
We will see that the net income at the comic store is $1,090, so the correct option is a.
What is Korey's net income?The net income will be the difference between how much money enters the store and how much he needs to pay.
We know that each month he gets $3,250.
And he must pay $1,285 for the comics and $875 in expenses, so the net income is:
N = $3,250 - $1,285 - $875 = $1,090
We conclude that the net income is $1,090, so the correct option is a.
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Solve the inequality. 2−6/5x≥−4
To solve the inequality 2 - (6/5)x >= -4, isolate x to find that x must be less than or equal to 5.
To solve the inequality 2 - (6/5)x >= -4, we first need to isolate the variable x. Let's start by adding (6/5)x to both sides of the inequality:
2 - (6/5)x + (6/5)x >= -4 + (6/5)xNext, we add 4 to both sides to get:
2 + 4 >= (6/5)xNow we divide both sides by (6/5) to solve for x:
5 >= xFinally, we can write the solution to the inequality:
x <= 5
So, x must be less than or equal to 5 to satisfy the inequality 2 - (6/5)x >= -4.
Solve. Alex borrowed $12.50 from his friend Danilo. He paid him back $8.75. How much does he still owe?
Another way to write 12 x 10 to the 3rd power
Which relationships would most likely be causal? Check all that apply.
a negative correlation between the temperature and the amount of snow still on the ground
a negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left
a positive correlation between the length of the side of a pool and its depth
a positive correlation between the height of a woman and the height of her brother
a negative correlation between the volume of water in a pot and the amount of time that the water takes to boil
Answer:
a negative correlation between the temperature and the amount of snow still on the ground a negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is leftStep-by-step explanation:
1. A negative correlation between the temperature and the amount of snow still on the ground
This is casual since temperature and amount of snow are inversely proportional to each other.
2. A negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left
This is casual since the number of digital photos uploaded and the amount of storage space are inversely proportional to each other.
3. A positive correlation between the length of the side of a pool and its depth
.
This is not casual since the length of the side of a pool and its depth are not related.
4. A positive correlation between the height of a woman and the height of her brother
.
This is not casual since the height of a woman and the height of her brother are not related.
5. A negative correlation between the volume of water in a pot and the amount of time that the water takes to boil
This is not casual since the volume of water in a pot and the amount of time that the water takes to boil are directly proportional.
A relationships would most likely be causal,
a negative correlation between the temperature and the amount of snow still on the ground.
a negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left
1. A negative correlation between the temperature and the amount of snow still on the ground
This is casual since temperature and amount of snow are inversely proportional to each other.
2. A negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left
This is casual since the number of digital photos uploaded and the
What is the relation of correlation?amount of storage space is inversely proportional to each other.
3. A positive correlation between the length of the side of a pool and its depth
This is not casual since the length of the side of a pool and its depth are not related.
4. A positive correlation between the height of a woman and the height of her brother
This is not casual since the height of a woman and the height of her brother are not related.
5. A negative correlation between the volume of water in a pot and the amount of time that the water takes to boil
This is not casual since the volume of water in a pot and the amount of time that the water takes to boil are directly proportional.
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Find the value of x. Round your answer to the nearest tenth. Show your work please!
A cook has 2 2/3 cups of flour. A recipe calls for 2 3/4 cups of flour. Does the cook have enough flour? If not, how much more flour is needed
A farmer wants to make a rectangular field with a total area of 800m2. It is surrounded by a fence. It is divided into 3 equal areas by fences. What is the shortest total length of fence with which this can be done?
To find the shortest total length of fence needed, divide the field into 3 equal areas. Use the area formula and differentiate to find the value of L that minimizes the perimeter. The shortest total length of fence needed is approximately 64.26m.
Explanation:To find the shortest total length of fence needed, we need to divide the field into 3 equal areas. Since the total area is 800m², each area will be 800m² ÷ 3 = 266.67m².
Now, let's find the dimensions of each area. Let the length of the rectangle be L and the width be W.
Since the areas are equal, we have LW = 266.67m². We are also given that the field is rectangular, so the perimeter is given by 2L + 2W.
To minimize the perimeter, we can use the formula 2L + 2W = P, and solve for L or W. Using the area formula and substituting, we have 2L + 2(266.67m²/L) = P.
Differentiating with respect to L and equating to zero, we can find the value of L that minimizes the perimeter. Solving the equation gives L ≈ 15.65m.
Therefore, the shortest total length of fence needed is approximately 2L + 2W ≈ 2(15.65m) + 2(266.67m²/15.65m) ≈ 64.26m.
For a rectangular field with a total area of 800 m², divided into three equal areas, the optimal solution is a square field with a side length of approximately 28.28 m. The shortest total length of fence needed is approximately 113.14 m.
Let's go through the step-by-step calculation:
1. **Given Information:**
- Total area of the rectangular field: [tex]\(800 \, \text{m}^2\).[/tex]
- The field is divided into 3 equal areas.
2. **Calculate the area of each section:**
[tex]\[ \text{Area of each section} = \frac{\text{Total area}}{\text{Number of sections}} = \frac{800 \, \text{m}^2}{3} \approx 266.67 \, \text{m}^2 \][/tex]
3. **Assume a square field:**
Let [tex]\(a\)[/tex] be the side length of the square field.
4. **Express the total area in terms of side length [tex](\(a\))[/tex]:**
[tex]\[ a^2 = 800 \, \text{m}^2 \][/tex]
5. **Calculate the side length [tex](\(a\))[/tex] :**
[tex]\[ a = \sqrt{800} \approx 28.28 \, \text{m} \][/tex]
6. **Calculate the total length of the fence (perimeter of the square):**
[tex]\[ \text{Perimeter} = 4a = 4 \times 28.28 \, \text{m} \approx 113.14 \, \text{m} \][/tex]
In a certain liberal arts college with about 10,000 students, 40% are males. if two students from this college are selected at random, what is the probability that they are of the same gender?
Probability lies at the heart of this mathematics question, involving the selection of two students at random from a college. The calculations involve determining the joint probability for the two students being of the same sex, either both male or both female, from the given student population distribution.
Explanation:The subject of this problem is probability. Here, in this liberal arts college, we've been given that 40% of 10,000 students, i.e., 4000 students are males; hence, 60% i.e., 6000 students are females.
When selecting two students at random, there are two possibilities: (1) both students are male or (2) both students are female.
The probability of both students being male can be calculated by multiplying: the probability of the first student being male (which is 4000/10000 = 0.4) by the probability of the second student also being male (which, after the first extraction, would be 3999/9999 since the total number of students and the number of male students both decrease by 1). Hence, the joint probability for two male students is: 0.4 * (3999/9999).
Similarly, we can calculate the joint probability for two female students using the same approach: 0.6 * (5999/9999).
The total probability that the two selected students are of the same gender would be the sum of these two probabilities.
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