Answer:
50 ft by 25 ft . . . . . 50 ft parallel to the barn
Step-by-step explanation:
You want the dimensions of the largest rectangular area that can be enclosed using 100 ft of fence for three sides.
PerimeterIf the dimensions of the space are L feet in length and W feet in width, where L is parallel to the barn, the length of the perimeter fence is ...
P = L +2W
Solving for W gives ...
W = (P -L)/2
AreaThe area of the enclosed space is ...
A = LW
A = L(P -L)/2 . . . . . substitute for W
Maximum areaThe area formula is the equation for a parabola that opens downward. It has zeros at L=0 and at L=P. The vertex (maximum) is found at the value of L that lies on the line of symmetry, halfway between these zeros. If we call that length M, then we have ...
M = (0 +P)/2 = P/2
The length of enclosure that maximizes the area is 1/2 the length of the available fence.
The width is ...
W = (P -P/2)/2 = P/4
The width of the enclosure that maximizes the area is 1/4 the length of the available fence.
Using 100 feet of fence, the dimensions are ...
length: 50 ft (parallel to the barn)width: 25 ft__
Additional comment
Note that we have solved this in a generic way. The solution given is the general solution to the 3-sided enclosure problem.
This is a special case of the general rectangular enclosure problem which has the solution that the total cost in one direction is equal to the total cost in the orthogonal direction. This rule applies even when costs are different for the different sides or for any partitions that might divide the enclosure.
Here the "cost" is simply the length of the fence. The 50 ft of fence parallel to the barn is equal in length to the 25 +25 ft of fence perpendicular to the barn.
The dimensions that give the maximum area inside the rectangle are x = 50 feet (parallel to the barn wall) and y = 25 feet (perpendicular to the barn wall).
To maximize the area of the rectangular fence, follow these steps:
1. Let's assign variables to the dimensions: let the length of the rectangle parallel to the barn wall be x feet, and the length perpendicular to the barn wall be y feet.
2. We are given that 100 feet of fence will be used for the other three sides. This means the fencing equation is:
x + 2y = 100.
3. Solve for x: x = 100 - 2y.
4. The area A of the rectangle is given by the product of its dimensions: A = xy.
5. Substitute the expression for x from step 3 into the area formula: A = (100 - 2y)y.
6. Expand the expression: A = 100y - 2y^2.
7. To maximize the area, we need to find the maximum value of the quadratic function A(y). Since the coefficient of the y^2 term is negative, the graph of A(y) is a downward-opening parabola, which means it has a maximum value.
8. To find the maximum, we'll use the vertex formula for parabolas: y_vertex = -b/(2a), where a = -2 and b = 100. Plugging in these values, we get y_vertex = -100/(2 * -2) = 25.
9. Substitute the value of y_vertex back into the equation for x: x = 100 - 2(25) = 50.
10. So the dimensions that give the maximum area inside the rectangle are x = 50 feet (parallel to the barn wall) and y = 25 feet (perpendicular to the barn wall).
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G is the incenter of ABC
GD:
BG:
FC:
BF:
For the given triangle when G is the incenter of ABC the value of DG = 12, FC = 35, BG = 13.41, BF = 6.
What is incenter?The intersection of all three of the triangle's interior angle bisectors is where a triangle's incenter is located. It may be thought of as the intersection of the triangle's internal angle bisectors. Due to the junction point of the central axis being the centre of the triangle's inscribed circle, this point will be equally spaced from each of the triangle's sides.
The centre of a triangle's inscribed circle, which is the biggest circle that can fit within the triangle, is known as the incenter.
Given that, G is the incenter of the triangle.
Thus, EG = GF = DG.
Now, the value of DG is determined using the Pythagoras Theorem:
(37)² = DG² + (35)²
1369 = DG² + 1225
DG = 12
Now, FC = 35.
For, BG we have:
BG² = 6² + 12²
BG² = 36 + 144
BG = 13.41
BF = 6
Hence, for the given triangle when G is the incenter of ABC the value of DG = 12, FC = 35, BG = 13.41, BF = 6.
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Which statements about the function y = x ^ 2 - 6x + 8 are true?
Answer:
There are multiple statements that can be made about the function y = x^2 - 6x + 8. Here are a few examples: 1. The function is a quadratic function, meaning it is a polynomial of degree 2. 2. The coefficient of the x^2 term is positive, which means the function opens upwards and has a minimum value. 3. The function intersects the y-axis at y = 8. 4. To find the x-coordinate of the vertex of the parabola, we can use the formula x = -b/(2a), where a is the coefficient of the x^2 term and b is the coefficient of the x term. In this case, a = 1 and b = -6, so x = 3. This means the vertex is at the point (3, 1). 5. The function can be factored
If you are going to draw four cards from a deck of cards, does drawing four aces from the deck have the same probability as drawing four 3s? Explain.
Answer:
No, drawing four aces from a deck of cards does not have the same probability as drawing four 3s.
Step-by-step explanation:
To see why, consider the number of ways to draw four aces from a deck of cards. There are only four aces in the deck, so the first card drawn must be an ace, which has a probability of 4/52 (or 1/13). After the first ace is drawn, there are only three aces left in the deck, so the probability of drawing a second ace is 3/51. Similarly, the probability of drawing a third ace is 2/50, and the probability of drawing the fourth ace is 1/49.
Therefore, the probability of drawing four aces from a deck of cards is:
(4/52) x (3/51) x (2/50) x (1/49) = 0.0000185
On the other hand, there are four 3s in the deck, so the first card drawn must be a 3, which has a probability of 4/52 (or 1/13). After the first 3 is drawn, there are only three 3s left in the deck, so the probability of drawing a second 3 is 3/51. Similarly, the probability of drawing a third 3 is 2/50, and the probability of drawing the fourth 3 is 1/49.
Therefore, the probability of drawing four 3s from a deck of cards is:
(4/52) x (3/51) x (2/50) x (1/49) = 0.0000185
The probabilities are the same for drawing four aces and drawing four 3s because the same process is followed for each scenario: a specific card is drawn, the number of remaining cards decreases by one, and the probability of drawing the next card changes accordingly.
In summary, the probability of drawing four aces from a deck of cards is the same as the probability of drawing four 3s, but they are both very low probabilities because they require drawing specific cards in a specific order from a large deck of cards.
i need help im confused
The parallelogram therefore has a 3/4 square inch area.
What is a parallelogram?A parallelogram is a geometric figure with four equal sides that are parallel to one another. It has equal opposed angles and two sets of parallel sides.
The base length and height of a parallelogram must be multiplied to determine its area. In this instance, the parallelogram's shorter side is specified as 1 inch and its longer side as 4 inches. It states that the height is 3/4 inch.
Hence, the parallelogram's area is:
Area = base x height
= 1 inch x 3/4 inch
= 3/4 square inches
3/4 square inches is equal to 1 inch by 3/4 inch by the base and height.
The parallelogram therefore has a 3/4 square inch area.
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Divide 2x^4-9x^3 5x^2 3x-8 by x^2-4x 1 and verify the division algorithms
Answer:
give up on math this just too hard
Step-by-step explanation:
no
Pls help ill give brainliest!
Answer:I only know that number 4 is yards i hope that is enough for now
Step-by-step explanation: in a game of soccer there are numbers that represent yards or any sport.
Step-by-step explanation:
a) The appropriate unit of measurement for the surface area of a tissue box could be square inches or square centimeters.
b) The appropriate unit of measurement for the amount of space in a gift box of earrings could be cubic inches or cubic centimeters.
c) The appropriate unit of measurement for the area of a parking lot could be square meters, square feet, depending on the size of the parking lot.
d) The appropriate unit of measurement for the length of a soccer field could be meters, yards or feet.
e) The appropriate unit of measurement for the volume of a medium-sized fish tank could be liters or cubic feet.
In the figure M and N are mid point at QP and PR prove that: respectively (1) MN = 1/2 QR →>> (ii) MN // QR M Q P N R
By the property of the mid point theorem of triangle:
(1) MN = 1/2 QR and (ii) MN || QR
Explain about the mid point theorem:A point that divides a line segment into two equal halves is the midway of the segment. The line segment's midpoint is located precisely in the middle.
According to the midpoint theorem, a segment is half the length of a triangle's third side if it is created by joining the midpoints of two of its sides.According to the midpoint theorem, a segment must be parallel toward the third side if it is created by joining the midpoints of two triangle's sides. Furthermore, it mentions that the length of that section is equal to that of the third side.(1) MN = 1/2 QR
By the property of the mid point theorem of triangle:
MN = 1/2 QR , As M and Na are the mid points sides QP and PR of triangle .
(ii) MN || QR
And thus, both the lines MN and QR will be parallel.
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complete question:
In the figure M and N are mid point at QP and PR prove that: respectively
(1) MN = 1/2 QR
(ii) MN || QR
The figure is attached.
You paid $16 in sales tax on a new phone. The phone costs $320. What is the percent of sales tax?
If you paid $16 in sales tax on a new phone. The phone costs $320. the percent of sales tax is 5%.
How to find the sales tax percentage?To find the percent of sales tax, we need to divide the amount of sales tax by the cost of the phone, and then multiply by 100 to convert to a percentage:
Percent of sales tax = (Sales tax / Cost of phone) x 100%
Percent of sales tax = (16 / 320) x 100%
Simplifying the expression inside the parentheses:
Percent of sales tax = 0.05 x 100%
Multiplying:
Percent of sales tax = 5%
Therefore, the percent of sales tax is 5%.
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There are 8,246 registered to vote in the town of mayfield. About how many people voted in the election? Explain(there are 74 voters)
We can estimate that about 74 people voted in the election, which is the same as the actual number of voters.
What are the voters?
To estimate the number of people who voted in the election, we can use the proportion of registered voters who actually voted.
If we know that there are 8,246 registered voters in the town of Mayfield and 74 voters cast their vote, we can calculate the proportion of registered voters who voted as follows:
Proportion of voters = Number of voters / Number of registered voters
Proportion of voters = 74 / 8,246
Proportion of voters = 0.00897
We can then use this proportion to estimate the number of people who voted by multiplying it by the total number of registered voters:
Estimated number of voters = Proportion of voters x Number of registered voters
Estimated number of voters = 0.00897 x 8,246
Estimated number of voters = 73.9 (rounded to the nearest whole number)
Therefore, we can estimate that about 74 people voted in the election, which is the same as the actual number of voters.
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Complete question is: There are 8,246 registered to vote in the town of mayfield, about 74 people voted in the election, which is the same as the actual number of voters.
PLS ANSWER! THIS WAS DUE SO ON MARCH 27TH AND THIS IS ONE OF THEM!
Answer:
444 in
Hope this helps!
Step-by-step explanation:
Area of a triangle is [tex]\frac{1}{2}[/tex] × B × H
Already given, the base is 12 and the height is 12.5, [tex]\frac{1}{2}[/tex] and 12 can simplify to get 6 so 6 × 12.5 is 75. One triangle has the area of 75, the 4 triangles have a total area of 300.
The base is a square, so the formula for that is L × W, which would be 12 × 12 = 144. Add that to 300 and the surface area of the pyramid is 444.
Which expression is equivalent to 4(15 - 7)?
The expression equivalent to 4(15 - 7) is 32.
What is an expression?
An expression is a combination of values, variables, operators, and function calls that are evaluated to produce a result. An expression can be as simple as a single value or as complex as a multi-level nested set of calculations.
We can simplify the expression 4(15 - 7) by performing the operation inside the parentheses first (since parentheses take precedence over multiplication). We get:
15 - 7 = 8
Substituting this value back into the expression, we get:
4(8)
Multiplying 4 by 8, we get:
32
Therefore, the expression equivalent to 4(15 - 7) is 32.
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Complete question is: The expression equivalent to 4(15 - 7) is 32.
Which statement about f(x) = -x squared + 8x-7 is true?
Option D : The statement that is true about the quadratic function f(x)=-x²+8x-7, which states that the zeros are 1 and 7, because f(x) = -(x - 1)(x - 7).
To find the zeros of the quadratic function f(x) = [tex]-x^2 + 8x - 7[/tex], we need to set f(x) equal to zero and solve for x:
[tex]-x^2 + 8x - 7 = 0[/tex]
We can solve this quadratic equation using the quadratic formula or by factoring. Let's use factoring.
We can rewrite the quadratic expression as:
[tex]-(x^2 - 8x + 7)[/tex]
To factor the expression, we need to find two numbers whose product is 7 and whose sum is -8. These numbers are -1 and -7. Therefore, we can rewrite the expression as:
-(x - 1)(x - 7)
So the zeros of the function are x = 1 and x = 7.
Therefore, the statement that is true is option D: The zeros are 1 and 7, because f(x) = -(x - 1)(x - 7).
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Which statement about f(x)=-x²+8x-7 is true?
A. The zeros are 1 and -7, because f(x)=-(x-1)(x+7)
B. The zeros are -1 and -7, because f(x)=-(x+1)(x+7)
C. The zeros are -1 and 7, because f(x)=-(x+1)(x-7)
D. The zeros are 1 and 7, because f(x)=-(x-1)(x-7)
Assume all lines that appear parallel, are parallel. Solve for x.
Value of variable x in the given triangle is 10 units.
Define triangle proportionalityTriangle proportionality refers to the relationship between the sides of two similar triangles. Similar triangles are those that have the same shape but may differ in size.
In similar triangles, the corresponding sides are in proportion, which means that the ratio of the lengths of any two corresponding sides is equal to the ratio of the lengths of any other pair of corresponding sides.
In the given figure;
Two lines of the triangle are parallel
By triangle proportionality rule,
AB/AC=AD/AE
7.6/11.6=19/19+x
7.6(19+x)=19×11.6
x=29-19
x=10
hence, value of variable x in the given triangle is 10 units.
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A farmer is painting a new barn. He will need to calculate the surface area of the barn to purchase the correct amount of paint. In which of the following units can the farmer expect to calculate the surface area?
a) m
b) m³
c) ft
d) ft²
The surface area of the square or other shaped barn will be in unit ft² i.e. D.
What are squares?
In mathematics, a square is a geometric shape that has four equal sides and four right angles. It is also known as a regular quadrilateral or a square polygon. The sides of a square are also called edges or vertices. The area of a square is given by the formula A = s², where s is the length of one of its sides.
Now,
Let the barn to be a square or any type of shape.
The farmer can expect to calculate the surface area of the barn in square units, such as square meters or square feet. This means that the correct option is d) ft².
Surface area is the measure of the total area that the surface of an object occupies. It is measured in square units because it involves calculating the area of multiple faces or surfaces of an object. In the case of the barn, the farmer will need to measure the area of each face of the barn that will be painted, such as the walls and roof, and add them together to get the total surface area. This calculation will give the farmer the total area of the barn that needs to be painted and the amount of paint needed to cover the area.
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Bailey and Terrance were scuba diving near a coral reef and timed their ascents back to the surface. The points in the table represent Bailey's depth (relative to sea level) after x minutes. Two times and depths for Terrance's ascent were recorded as points. Assume the points lie on a line. Which diver ascended at a faster rate? How much faster did that diver ascend?
We can follow these steps using the points provided in the table and for Terrance to determine which diver ascended at a faster rate and by how much.
To determine which diver ascended at a faster rate, we need to calculate the slope of the line for each diver's ascent. The slope represents the rate of change in depth over time.
A steeper slope indicates a faster ascent.
Step 1: Find the slope of Bailey's ascent using the points given in the table.
we can use the slope formula:
(y2 - y1) / (x2 - x1).
Step 2: Find the slope of Terrance's ascent using the two points given for Terrance.
Step 3: Compare the slopes of both divers.
The diver with the steeper slope (greater slope value) ascended at a faster rate.
Step 4: To determine how much faster the faster diver ascended, subtract the smaller slope value from the larger slope value.
Without the actual data points, I cannot provide specific calculations.
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Struggling with this question as well. I suck at maths!
Answer:
Yes, the triangle with side lengths 6, 8, and 10 is a right triangle.
Explaination:
We can use the Pythagorean theorem to check if a triangle is a right triangle or not. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side (the hypotenuse). In other words:
a^2 + b^2 = c^2
where a and b are the lengths of the two shorter sides, and c is the length of the longest side (the hypotenuse).
In this case, the two shorter sides are 6 and 8, and the longest side (the hypotenuse) is 10. So we can plug these values into the equation above:
6^2 + 8^2 = 10^2
Simplifying:
36 + 64 = 100
100 = 100
Since the equation is true, this means that the triangle satisfies the Pythagorean theorem, and therefore it is a right triangle.
In order to solve the equation 9x=−3(x+1/3y) for y , a student simplifies the equation as shown: Step 1: 9x=−3x+−3(1/3y) Step 2: 9x=−3x+−y Step 3: 9x−3x=−y Step 4: 6x=−y Step 5: y=−6x Which statement describes the error the student made in simplifying the equation?
The error the student made was not correctly distributing the -3 to the second term (1/3y) inside the parenthesis in Step 1. The student incorrectly wrote -3(1/3y) as -3y instead of -y.
By correctly distributing the -3, the proper simplification would lead to the final answer of y = -12x.
The error the student made in simplifying the equation occurred in Step 1.
To solve the equation 9x = -3(x + 1/3y) for y is as follows:
Step 1: Distribute the -3 to both terms inside the parenthesis:
9x = -3x - y
Step 2: Add 3x to both sides of the equation to isolate the term with y:
9x + 3x = -y
Step 3: Simplify the equation by combining like terms:
12x = -y
Step 4: To solve for y, multiply both sides of the equation by -1:
y = -12x.
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x^(2)-10x-7 in vertex form
Answer:
y= 2 (x - -5/2)^2-39/2 ?
a triangular prism has a volume of 27.36 cubic units.the prism is 2.88units tall. what is the area of the triangular base
The area of the triangular base of the prism is 9.5 unit².
What is an area?
To find the area of the triangular base, we need to use the formula for the volume of a triangular prism:
V = (1/2)×b×h×H
Where b is the length of the base of the triangle, h is the height of the triangle, and H is the height of the prism.
We know that the volume of the prism is 27.36 unit³ and the height of the prism is 2.88 units. So we can plug these values into the formula:
27.36 = 1/2×b×h×2.88
Simplifying this equation, we get:
b×h = 19
We don't know the values of b and h separately, but we can use the fact that the area of a triangle is equal to (1/2)×b×h. So we can rewrite the equation as:
2 × (Area of triangular base) = 19
Dividing both sides by 2, we get:
Area of triangular base = 9.5 units²
Therefore, the area of the triangular base of the prism is 9.5 unit².
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focus groups of 10 people are randomly selected to discuss products of the famous company. it is determined that the mean number (per group) who recognize the famous brand name is 4.80000019, and the standard deviation is 0.73. would it be unusual to randomly select 10 people and find that greater than 8 recognize the famous brand name? group of answer choices
Yes 10 people would be unusual as it falls outside of 2 standard deviation of mean.
Here, the mean number of the famous brand name is 4.80000019, and the standard deviation is 0.73
so, μ = 4.80000019 and σ = 0.73
the number of people = 10
⇒ n = 10
Consider μ ± 2σ
= 4.80000019 ± 2(0.73)
= 4.80000019 ± 1.46
= [4.80000019 - 1.46, 4.80000019 + 1.46]
= [3.34000019, 6.26000019]
As we know 6.26000019 < 8
This means that, it falls outside of 2 standard deviation of mean.
Therefore, it would be unusual to randomly select 10 people and find that greater than 8 recognize the famous brand name.
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This season, the probability that the Yankees will win a game is 0.51 and the probability that the Yankees will score 5 or more runs in a game is 0.59. The probability that the Yankees lose and score fewer than 5 runs is 0.33. What is the probability that the Yankees will lose when they score fewer than 5 runs? Round your answer to the nearest thousandth.
The probability that the Yankees will lose when they score less than 5 runs is 0.805.
What exactly is probability?
Probability is a measure of the possibility of an event to be occurred. It is a numerical value between 0 and 1, with 0 indicating that the event is impossible, and 1 indicating that the event is certain.
Now,
Let's denote the event "Yankees win a game" by W, and the event "Yankees score 5 or more runs in a game" by S. Then, we are given:
P(W) = 0.51
P(S) = 0.59
P(W' and S') = 0.33, where W' denotes the complement of the event W (i.e., Yankees lose a game), and S' denotes the complement of the event S (i.e., Yankees score fewer than 5 runs).
We can use the following formula to find the probability that the Yankees lose when they score fewer than 5 runs:
P(W' | S') = P(W' and S') / P(S')
Using the given values, we have:
P(W' | S') = 0.33 / (1 - 0.59)
= 0.33 / 0.41
≈ 0.805
Rounding to the nearest thousandth, we get:
P(W' | S') ≈ 0.805
Therefore, the probability that the Yankees will lose when they score fewer than 5 runs is approximately 0.805.
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Problems 15 & 16. Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar. If the triangles are similar, write a valid similarity statement.
Part A : From Angle Angle similarity, ΔMBR and ΔZLP are similar.
Part B.: As the ratios of sides are not equal, triangles are not similar.
Explain about the similar triangles:Congruent triangles and similar triangles are two different but closely related concepts. Similar triangles are any two or more triangles having the identical proportions of their corresponding sides, an equal pair of corresponding angles, and their similar shape.
Part A:
using the equal sides property of triangle:
In ΔRBM,
side BM = side RM
So,
angle B = angle R (eq 1)
In triangle ZLP and PL = LZ
So, angle P = angle Z (eq 2)
And angle B = angle P
∠B = ∠R = ∠P = ∠Z
By Angle Angle similarity, triangle RBM and triangle ZLP are similar.
Part 16.:
In right angle triangle: KLN.
(KL)² = LN² + KN²
(KL)² = 8² + 15²
(KL)² = 64 + 225
(KL)² = 289
(KL) = 17
taking the ratios of sides:
27/8 = 30.6/17
3.375 ≠ 1.8
As the ratios of sides are not equal, triangles are not similar.
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find the value of x.
Write the quadratic function in vertex form that has a
vertex of (2, 1) and goes through the point (4, 5)
Answer:
Sure, I can help you with that! First, we know that the vertex form of a quadratic function is: f(x) = a(x - h)^2 + k where (h, k) is the vertex of the parabola. So we can plug in the values we have: f(x) = a(x - 2)^2 + 1 Now we just need to find the value of "a". To do that, we can use the fact that the function goes through the point (4, 5). So we can plug in those values: 5 = a(4 - 2)^2 + 1 Simplifying, we get: 5 = 4a + 1 4a = 4 a = 1 So now we know that the quadratic function is: f(x) = (
A chef has 1834
pounds of shrimp to make gumbo.
Part A
Each batch of gumbo requires 334
pounds of shrimp. Which solution shows how many batches of gumbo can be made?
70 batches; 1834×334
15
batch; 334÷1834
6 batches; 1834÷334
5 batches; 1834÷334
Part B
The chef could make a larger batch of gumbo that uses 614
pounds of shrimp.
How many fewer larger batches of gumbo can be made than smaller batches?
Answer:
1834 ÷ 334 ≈ 5.49
Since we can't make a fraction of a batch, we can make 5 batches of gumbo. therefore, the solution is
5 batches: 1834÷334
Part B:
For the smaller batches, each batch requires 334 pounds of shrimp. For the larger batches, each batch requires 614 pounds of shrimp.
subtract
614-334 = 280 pounds
divide
1834÷614 ≈ 2.99
Since we can't make a fraction of a batch, we can make 2 larger batches of gumbo.
So, the solution is: 2 fewer larger batches can be made than smaller batches.
evaluate the question when m=8 m^2 + 9
Here is your answer hope this helps:
m=8
8x8=64
64+9= 73
Please some one help and including answer this is due today hurry please.
To find the minimum value of a quadratic function most efficiently, we would like to see the quadratic function in its vertex form.
How to solveThe vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola, and "a" determines the direction and the "stretch" of the parabola.
If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.
The minimum value of the function occurs at the vertex if the parabola opens upwards (a > 0), and the maximum value occurs at the vertex if the parabola opens downwards (a < 0).
In this case, since we're looking for the minimum value, we would want a quadratic function where a > 0. Once we have the vertex form, we can easily identify the vertex (h, k), and the minimum value of the function will be k.
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Write (9m)⁴ without exponents.
we can multiply 6561m⁴ by 1 to get the final answer of 6561m⁴.
How to solve the problem?
In mathematics, an exponent refers to a numerical representation of repeated multiplication. An exponent is usually written as a small number that appears to the right and above a larger number, called the base. The exponent indicates how many times the base should be multiplied by itself. For example, in the expression 3², the base is 3, and the exponent is 2. This means that 3 should be multiplied by itself 2 times, resulting in 3 multiplied by 3, which equals 9.
Exponents are a shorthand way of representing repeated multiplication and can be used to simplify complicated expressions. They are widely used in algebra, calculus, and other branches of mathematics. Exponents can be positive, negative, or zero, and they can be fractions or irrational numbers. The rules of exponents, such as the product rule and power rule, allow for easy manipulation of expressions involving exponents.
To write (9m)⁴ without exponents, we need to perform the multiplication four times.
First, we can multiply 9m by 9m to get 81m².
Next, we can multiply 81m² by 9m to get 729m³.
Then, we can multiply 729m³ by 9m to get 6561m⁴.
Finally, we can multiply 6561m⁴ by 1 to get the final answer of 6561m⁴.
In summary, (9m)⁴ can be written as 6561m⁴ by performing multiplication four times using the distributive property of multiplication. It is important to note that exponents are a shorthand way of writing repeated multiplication, but the same result can always be obtained by performing the multiplication explicitly. This process may become more cumbersome for larger exponents, which is why exponents are a useful notation for expressing repeated multiplication in a more concise way.
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Sylvia earns commission on insurance sales. She sold a plan that costs 6,000 per year. Based on the chart, how much commission will Sylvia earn from this sale over the first three years?
The total commission Sylvia will earn over the first three years is $1,500.
Let's assume the commission rates are X% for the first year, Y% for the second year, and Z% for the third year.
Calculate the yearly commission amounts
To calculate Sylvia's yearly commission, we'll multiply the insurance plan cost (6,000) by the respective commission rate for each year.
First-year commission = 6,000 * (X / 100)
Second-year commission = 6,000 * (Y / 100)
Third-year commission = 6,000 * (Z / 100).
Calculate the total commission over three years
To find the total commission Sylvia will earn over the first three years, we'll add the yearly commission amounts calculated.
Total commission = First-year commission + Second-year commission + Third-year commission.
To calculate the commission Sylvia will earn over the first three years, we need to use the commission rates for each year:
In the first year, Sylvia will earn a 10% commission on the $6,000 plan, which is $600.
In the second year, Sylvia will earn an 8% commission on the $6,000 plan, which is $480.
In the third year, Sylvia will earn a 7% commission on the $6,000 plan, which is $420.
Therefore, the total commission Sylvia will earn over the first three years is:
$600 + $480 + $420 = $1,500
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Question: Sylvia earns commission on insurance sales. She sold a plan that costs 6,000 per year. Based on the chart, how much commission will Sylvia earn from this sale over the first three years?
Commission Schedule.
Year Rate
1 st 10
2 nd 8%
3 rd 7%
Mrs. Harper is teaching a 5th grade class. She is
standing 6 meters in front of Annie. Javier is
sitting to Annie's right. If Javier and Mrs. Harper
are 8 meters apart, how far apart are Annie and
Javier? If necessary, round to the nearest tenth.
This is a Pythagorean theorem word problem
Answer:
Annie and Javier are 10 meters apart.
Step-by-step explanation:
We can use the Pythagorean theorem to solve this problem. Let's call the distance between Annie and Javier "x". Then, we have two right triangles:
The first right triangle is formed by Mrs. Harper, Annie, and the distance between Mrs. Harper and Annie. We know that Mrs. Harper is 6 meters in front of Annie, so the hypotenuse of this triangle is 6 meters.
The second right triangle is formed by Mrs. Harper, Javier, and the distance between Mrs. Harper and Javier. We know that Mrs. Harper and Javier are 8 meters apart, so the hypotenuse of this triangle is 8 meters.
Since Mrs. Harper is the common point of both triangles, we can combine them to form a larger right triangle with sides of 6 meters, 8 meters, and x meters:
Using the Pythagorean theorem, we have:
6^2 + 8^2 = x^2
Simplifying:
36 + 64 = x^2
100 = x^2
Taking the square root of both sides:
x = 10
Therefore, Annie and Javier are 10 meters apart.