Consider the function f(x) = x1/3(x + 2)2/3 whose domain is (−[infinity], [infinity]).(a)the intervals on which f is increasing. (Enter your answer as a comma-separated list of intervals.) (0, 2^(3/2)) the intervals on which f is decreasing. (Enter your answer as a comma-separated list of intervals.) is (-∞, 0) and (2^(3/2), ∞). the open intervals on which f is concave up. (Enter your answer as a comma-separated list of intervals.) (-∞, (5/4)^(3/2)) the open intervals on which f is concave down is ((5/4)^(3/2), ∞). the local extreme values of f. (Round your answers to three decimal places.)local minimum value is 0, local maximum value 4(2)^(1/3). the global extreme values of f on the closed, bounded interval [−3, 2].(Round your answers to three decimal places.) global minimum value -1.442 global maximum value 4(2)^(1/3).
To find the intervals on which f(x) is increasing or decreasing, we need to find the derivative of f(x) and determine its sign.
f(x) = x^(1/3)(x + 2)^(2/3)
f'(x) = (1/3)x^(-2/3)(x + 2)^(2/3) + (2/3)x^(1/3)(x + 2)^(-1/3)
Simplifying f'(x), we get
f'(x) = (x + 2)^(1/3)[2 - x^(2/3)] / (3x^(2/3))
The derivative is defined for all x, except for x = 0.
To find the intervals on which f(x) is increasing or decreasing, we need to find the values of x that make f'(x) equal to zero or undefined.
Setting f'(x) = 0, we get
2 - x^(2/3) = 0
x = 2^(3/2)
Setting the denominator of f'(x) equal to zero, we get
x = 0
Thus, the critical points of f(x) are x = 0 and x = 2^(3/2).
We can now use these critical points to determine the intervals on which f(x) is increasing or decreasing.
When x < 0, f'(x) < 0, so the function is decreasing.
When 0 < x < 2^(3/2), f'(x) > 0, so the function is increasing.
When x > 2^(3/2), f'(x) < 0, so the function is decreasing.
Therefore, f(x) is increasing on the interval (0, 2^(3/2)) and decreasing on (-∞, 0) and (2^(3/2), ∞).
To find the intervals on which f(x) is concave up or concave down, we need to find the second derivative of f(x) and determine its sign.
f''(x) = (2/9)x^(-5/3)(x + 2)^(2/3)[4 - x^(2/3)]
Simplifying f''(x), we get
f''(x) = (2/9)(x + 2)^(1/3)(4x^(2/3) - 5) / x^(5/3)
The second derivative is undefined at x = 0 and x = -2.
Setting f''(x) = 0, we get:
4x^(2/3) - 5 = 0
x = (5/4)^(3/2)
Using these critical points, we can determine the intervals on which f(x) is concave up or concave down.
When x < (5/4)^(3/2), f''(x) > 0, so the function is concave up.
When x > (5/4)^(3/2), f''(x) < 0, so the function is concave down.
Therefore, f(x) is concave up on (-∞, (5/4)^(3/2)) and concave down on ((5/4)^(3/2), ∞).
To find the local extreme values of f(x), we need to examine the critical points.
At x = 0, f(x) = 0.
At x = 2^(3/2), f(x) = 4(2)^(1/3).
Therefore, the local minimum value of f(x) is 0, which occurs at x = 0, and the local maximum value of f(x) is 4(2)^(1/3), which occurs at x = 2^(3/2).
To find the global extreme values of f(x) on the closed, bounded interval [-3, 2], we need to examine the critical points and the endpoints of the interval.
At x = -3, f(x) = (-1)^(1/3), which is approximately -1.442.
At x = 0, f(x) = 0.
At x = 2^(3/2), f(x) = 4(2)^(1/3).
At x = 2, f(x) = 2^(2/3)(4)^(2/3) = 4(2)^(1/3).
Thus, the global minimum value of f(x) on the interval [-3, 2] is approximately -1.442, which occurs at x = -3, and the global maximum value of f(x) on the interval [-3, 2] is 4(2)^(1/3), which occurs at x = 2^(3/2).
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Help i don't understand this pls help
The calculated height of the pyramid is 8.49 cm
Calculating the height of the pyramidFrom the question, we have the following parameters that can be used in our computation:
Pyramid = square based
base lengths = 14 cm
Slant height = 11 cm
The height (h) of the pyramid can be calculated using
h² = Slant height² - (base lengths/2)²
Substitute the known values in the above equation, so, we have the following representation
h² = 11² - (14/2)²
So, we have
h² = 72
Take the square roots
h = √72
So, we have
h = 8.49 cm
Hence, the height of the pyramid is 8.49 cm
The sketch is added as an attachment
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Help! Multiply (3.25 x 10−9)(6.7 x 1015). Write the final answer in scientific notation.
21.775 x 10−135
2.1775 x 10−134
21.775 x 106
2.1775 x 107
Answer:
D. 2.1775 x 10^7
Step-by-step explanation:
I just finished the quiz and got it correct
Hope its not to late!
Answer:
2.1775 x 107
Step-by-step explanation:
PLEASE helppppp!!!
A group of students and teachers are going on a field trip. One-ninth of the group will fit on 1/3 of a school bus. How many buses are needed to transport the entire group?
Answer:
3 buses
Step-by-step explanation:
Total number of students=1
1/9 =1/3of the school bus
1=?
(1*1/3)÷(1/9)= 3 buses
A bleach and water solution with a 2:3 ratio means: A 1/3 part bleach and 2/3 part water B 2 cups of bleach and 3 cups of water C 3 cups of bleach and 2 cups of water
The correct interpretation of a bleach and water solution with a 2:3 ratio would be option B: 2 cups of bleach and 3 cups of water.
A bleach and water solution with a 2:3 ratio means that for every 2 parts of bleach, there should be 3 parts of water. This ratio is typically expressed in terms of volume or quantity.
To understand this ratio, let's break it down using different units:
A. 1/3 part bleach and 2/3 part water:
If we consider 1/3 part bleach, it means that for every 1 unit of bleach, there should be 2 units of water. However, this does not match the given 2:3 ratio.
B. 2 cups of bleach and 3 cups of water:
If we consider cups as the unit of measurement, this means that for every 2 cups of bleach, there should be 3 cups of water. This matches the given 2:3 ratio, making it a valid interpretation.
C. 3 cups of bleach and 2 cups of water:
If we consider cups as the unit of measurement, this means that for every 3 cups of bleach, there should be 2 cups of water. However, this interpretation does not match the given 2:3 ratio.
Based on the given options, the correct interpretation of a bleach and water solution with a 2:3 ratio would be option B: 2 cups of bleach and 3 cups of water.
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the result of a number when increased by 15% is 161. Find the number
Answer:
(15×7)×29/€549×898×68
What do i do i didn’t understand?
Answer:
-1
Step-by-step explanation:
substitute (8-3) / -5
5 / -5
-1
A school is building a new playground. They have spent 75% of the money for the new playground. If they have spent $1200, what is the total amount of money they will spend to build the playground? ASAP
find the area of the rectangle
\(A = (a^2 b)(3 a^2 b) = 3(a^2 b)^2 = 3a^4 b^2 ~~ \text{units}^2,\)
Answer:
3a⁴b² units²
Step-by-step explanation:
Area of rectangle: L x B
=> Area of rectangle: LB
=> Area of rectangle = a²b x 3a²b
=> Area of rectangle = a² x b x 3 x a² x b
=> Area of rectangle = a² x b x 3 x a² x b
=> Area of rectangle = a² x a² x b x b x 3
=> Area of rectangle = a⁴ x b² x 3
Therefore, the area of the rectangle is 3a⁴b² units²
Hoped this helped.
Math pls help for 26 points
Answer:
77
Step-by-step explanation:
These cones are similar. Find the volume of the smaller cone. Round to the nearest tenth. 2 cm 5 cm Volume=[?]cm^ 3 Volume me = 131 cm ^ 3
Answer:
8.4 cm^3 to the nearest tenth.
Step-by-step explanation:
The ratio of the volumes of the 2 cones is the ratio of the cubes of their radii.
So:
5^3 / 2^3 = 131 / v
v = 2^3 * 131 / 5^3
= 8 * 131 / 125
= 8.384.
The volume of smaller cone is 8.4 cm³
What is Volume of Cone?The volume of a cone formula is given as one-third the product of the area of the circular base and the height of the cone. According to the geometric and mathematical concepts, a cone can be termed as a pyramid with a circular cross-section. By measuring the height and radius of a cone, you can easily find out the volume of a cone. If the radius of the base of the cone is "r" and the height of the cone is "h", the volume of cone is given as V = (1/3)πr²h.
By applying Pythagoras theorem on the cone, we can find the relation between volume and slant height of the cone.
We know, h² + r² = L²
⇒ h = √(L² - r²)
where,
h is the height of the cone,
r is the radius of the base, and,
L is the slant height of the cone.
The volume of the cone in terms of slant height can be given as V = (1/3)πr²h = (1/3)πr²√(L² - r²).
Given:
The ratio of the volumes of the 2 cones is the ratio of the cubes of their radii.
Then,
5³ / 2³ = 131 / v
v = 2³ * 131 / 5³
v = 8 * 131 / 125
v = 8.384 cm³
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Which shows how to find the value of this expression when x = negative 2 and y = 5?.
The value of this expression is 576/365
What is Expression?
A phrase, also known as an expression in some circumstances, is a group of words or a single word that functions as a grammatical unit. For instance, the English word "the very joyful squirrel" is a noun phrase which contains the adjective phrase "extremely happy". Phrases can be made up of a single word or a whole sentence.
(3x³y⁻²)² = 576/365
Given,
x = -2
y = 5
Applying property of exponents to simplify the expression.
Property: \((a^{b} )^{c}\) =\(a^{bc}\)
So, we have : (3x³y⁻²)²
= 3²x⁶y⁻⁴
= 9x⁶y⁻¹
Property : a⁻ᵇ
= 1/aᵇ
= 9x⁶ / y⁴
Now, plugging in values of x and y
= 9 (-2)⁶ / 5⁴
= 9*64/625
= 576 / 365
This phrase has a value of 576/365.
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The value of this expression is 576/365
What is Expression?
A phrase, also known as an expression in some circumstances, is a group of words or a single word that functions as a grammatical unit. For instance, the English word "the very joyful squirrel" is a noun phrase which contains the adjective phrase "extremely happy". Phrases can be made up of a single word or a whole sentence.
(3x³y⁻²)² = 576/365
Given,
x = -2
y = 5
Applying property of exponents to simplify the expression.
So, we have : (3x³y⁻²)²
= 3²x⁶y⁻⁴
= 9x⁶y⁻¹
Property : a⁻ᵇ
= 1/aᵇ
= 9x⁶ / y⁴
Now, plugging in values of x and y
= 9 (-2)⁶ / 5⁴
= 9*64/625
= 576 / 365
This phrase has a value of 576/365.
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m
10) Skylar claims that rigid transformations were used to map AB to A'B', for A(-1,13
B(1,5), A '(2, 3), and B'(6, 1). Is Skylar correct?
Yes; the distance from A to A'is the same as the distance from B to B’
No; the measure of AB is less than the measure of A'B'.
Yes; the measure of AB is the same as the measure of A'B'
No; the measure of AB is greater than the measure of A'B'.
The transformation used to map AB to A'B' is not a rigid transformation since the measure of AB is greater than the measure of A'B' and not equal.
What is transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, rotation, translation and dilation.
A rigid transformation is a transformation that preserves the shape and size such as rotation, reflection and translation.
Given the points A(-1, 13), B(1, 5), A'(2, 3), and B'(6, 1), hence:
\(AB=\sqrt{(5-13)^2+(1-(-1))^2} =\sqrt{68}\\\\A'B'=\sqrt{(1-3)^2+(6-2)^2}=\sqrt{20}\)
The transformation used to map AB to A'B' is not a rigid transformation since the measure of AB is greater than the measure of A'B' and not equal.
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A cylinder has a radius of 10 yards. Its volume is 3,768 cubic yards. What is the height of the cylinder?
Use ≈ 3.14 and round your answer to the nearest hundredth.
h ≈
yards what would the answer be
Rounded to the nearest hundredth, the height of the cylinder is approximately 11.98 yards.
What is volume?In mathematics, volume is a quantity that measures the amount of space occupied by an object or a shape. It is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet. The volume of an object can be calculated using its dimensions, such as length, width, and height, or using mathematical formulas specific to the shape of the object, such as the formula for the volume of a cylinder or sphere.
Here,
The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height. We are given that the radius is 10 yards and the volume is 3,768 cubic yards. So we can set up the equation:
3,768 = π(10)²h
Simplifying and solving for h, we get:
h = 3,768 / (π(10)²)
h ≈ 11.98 yards
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Daniel began filling an empty swimming pool at 8:30 a.M. After 5 hours, the pool contained 230 gallons of water. If the water input contained at a constant rate, how many more hours are needed to completely fill the 920-gallon pooo?
It will take 20 hours to fill the 920-gallon water in the pool if Daniel began filling an empty swimming pool at 8:30 a.M. After 5 hours, the pool contained 230 gallons of water.
What is a fraction?Fraction number consists of two parts one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
We have:
After 5 hours the pool filled with 230 gallons
Let's suppose that after x hours the pool will fill with 920 gallons
After x hours the pool filled with 920 gallons.
As the water input contained at a constant rate
So the value of x can be evaluated:
\(\rm x = \frac{920\times5}{230}\)
x = 20 hours
Thus, it will take 20 hours to fill the 920-gallon water in the pool if Daniel began filling an empty swimming pool at 8:30 a.M. After 5 hours, the pool contained 230 gallons of water.
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llow
1. Find the average rate of change tor the
Explain or show your reasoning.
a. between noon and 1 p.m.
b. between noon and 4 p.m.
c. between noon and midnight
The average rate of change for the given intervals are as follows:
Determine the rate of change?a) Between noon and 1 p.m.
To calculate the average rate of change between noon and 1 p.m., we need to determine the difference in the value of the quantity of interest at these two time points and divide it by the duration of the interval. Without specific information about the quantity or function being considered, it is not possible to determine the average rate of change.
The average rate of change depends on the specific function or quantity and its behavior over the given time interval.
b) Between noon and 4 p.m.
Similarly, without specific information about the quantity or function, it is not possible to determine the average rate of change between noon and 4 p.m. The average rate of change depends on the specific function or quantity and its behavior over the given time interval.
c) Between noon and midnight
Again, without specific information about the quantity or function, it is not possible to determine the average rate of change between noon and midnight. The average rate of change depends on the specific function or quantity and its behavior over the given time interval.
In summary, the average rate of change cannot be determined without additional information about the specific function or quantity being considered.
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Aaron used the pythagorean theorem to find the height of a tree. he calculated that the tree was square root of 625 feet tall. which of these following should be used to write the height of the tree?
The height of the tree should be written as 25 feet.
If Aaron used the Pythagorean theorem to find the height of a tree and obtained the result as the square root of 625 feet, we need to simplify the square root expression to find the actual height of the tree.
The square root of 625 is a mathematical operation that asks "What number, when multiplied by itself, gives the result of 625?" In this case, the square root of 625 is 25 because 25 * 25 = 625.
Therefore, the height of the tree should be written as 25 feet. This means that Aaron determined the height of the tree to be 25 feet using the Pythagorean theorem.
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(02.03 M)
Write the equation of a function whose parent function, f(x) = x + 6, is shifted 4 units to the right.
Answer:
\(\boxed{f(x) = x + 2}\)
Step-by-step explanation:
The translation is horizontal translation.
The value of x is subtracted from 4, since the function is shifted 4 units to the right.
f(x - 4)
f(x) = (x - 4) + 6
f(x) = x + 2
Which type of construction is illustrated in the figure?
Answer:
i think its c
Step-by-step explanation:
Answer:
The bisector of a given angle.
Step-by-step explanation: the correct answer is the first one because it is giving you an angle and if you didn't know what a bisector is it is a line that bisects an angle meaning that each side of the line is equal to one another.
Eleanor must earn at least 70 points for her book report project. So far on the rubric she has 54 points. Write and solve an inequality to show how many points Madeline needs.
Answer:
\(x + 54\ge 70\)
\(x \ge 16\)
Step-by-step explanation:
Given
\(Total = 70\) (at least)
\(Accumulated = 54\)
Required
Express as an inequality
In inequalities, at least means \(\ge\)
So, the expression is:
\(x + Accumulated \ge Total\)
Where x represents the additional points
\(x + 54\ge 70\)
Solve for x
\(x \ge 70 - 54\)
\(x \ge 16\)
Hence, she needs at least 16 points
-3x+7y=5x+2y complete the missing value in the solution to the equation
a triangle has two sides of length 18 and 3. what is the largest possible whole-number length for the third side?
Answer:
20
Step-by-step explanation:
Triangle inequality says that the third side can only be 18-3...15, that is bigger than 15.
And 18+3... 21, that is smaller than 21.
If the third side is 21, the 18 and the 3 will just lay right on top of the 21 and not make a triangle. So it has to be 20 in order to be a whole number.
G(Q) = 5 + 3Q + 202 - Q2 C2(Q) = 3 + 4Q + 2 1. Find the MC function for both C1(Q) AND C2(Q). 2. Find AVC function for both Ci(Q) AND C2(Q). 3. Find AFC function for both C1(Q) AND C2(Q). 4. Find AC function for both Ci(Q) AND C2(Q). 5. Find ATC function for both Ci(Q) AND C2(Q).
For C1(Q) = 3 - 2Q.
For C2(Q) = 4.
2. The AVC function
For C1(Q) = 5/Q + 3 + 20/Q - Q.
For C2(Q) = 3/Q + 4 + 2/Q.
3. The AFC function
For C1(Q)= 5/Q - 20/(5 + 3Q + 20/Q - Q)
For C2(Q) = 0.
4. To find the AC function
For C1(Q) = (5 + 3Q + 202 - Q^2)/Q + 5/Q - 20/(5 + 3Q + 20/Q - Q).
For C2(Q) = (3 + 4Q + 2)/Q + 3/Q + 4 + 2/Q.
5.To find the ATC function
For C1(Q)= 5/Q² + 3/Q + 20/Q² - Q/Q + 5/Q - 20/(5Q + 3Q² + 20 - Q²)
For C2(Q)= 3/Q² + 4/Q + 2/Q² + 3/Q + 4/Q + 2/Q.
Find the ATC functions for C1(Q) and C2(Q) given the provided cost functions?
1. To find the MC function, we take the derivative of the cost functions with respect to Q.
For C1(Q) = 5 + 3Q + 202 - Q^2, MC1(Q) = 3 - 2Q.
For C2(Q) = 3 + 4Q + 2, MC2(Q) = 4.
2. To find the AVC function, we divide the cost functions by Q.
For C1(Q), AVC1(Q) = (5 + 3Q + 202 - Q^2)/Q = 5/Q + 3 + 20/Q - Q.
For C2(Q), AVC2(Q) = (3 + 4Q + 2)/Q = 3/Q + 4 + 2/Q.
3. To find the AFC function, we subtract the AVC function from the ATC function.
For C1(Q), AFC1(Q) = (5 + 3Q + 202 - Q^2)/Q - (5 + 3Q + 202 - Q^2)/(5 + 3Q + 20/Q - Q)
= 5/Q - 20/(5 + 3Q + 20/Q - Q).
For
C2(Q), AFC2(Q) = (3 + 4Q + 2)/Q - (3 + 4Q + 2)/(3/Q + 4 + 2/Q) = 0.
4. To find the AC function, we add the AVC function to the AFC function.
For
C1(Q), AC1(Q) = (5 + 3Q + 202 - Q^2)/Q + 5/Q - 20/(5 + 3Q + 20/Q - Q).
For
C2(Q), AC2(Q) = (3 + 4Q + 2)/Q + 3/Q + 4 + 2/Q.
5. To find the ATC function, we divide the AC function by Q.
For
C1(Q), ATC1(Q) = [(5 + 3Q + 202 - Q²)/Q + 5/Q - 20/(5 + 3Q + 20/Q - Q)]/Q
= 5/Q² + 3/Q + 20/Q² - Q/Q + 5/Q - 20/(5Q + 3Q² + 20 - Q²).
For
C2(Q), ATC2(Q) = [(3 + 4Q + 2)/Q + 3/Q + 4 + 2/Q]/Q
= 3/Q² + 4/Q + 2/Q² + 3/Q + 4/Q + 2/Q.
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How many terms are in the expression 3x+y−23−5
Answer:
There are 4 terms
Step-by-step explanation:
“Terms are single numbers, variables, or the product of a number and variables. Examples of terms: 9 a 9a 9a. y y y.”
Solve the system of linear equations by adding or subtracting. 2x +y = -6 -5x + y = 8 Enter the solution as an ordered pair. The solution is
Answer: (-6,-4)
Step-by-step explanation:
jenn randomly arranges the three digits $1$, $2$, and $3$ to create a three-digit number. what is the probability that her number is a multiple of $4$?
To find the probability that Jenn's three-digit number is a multiple of 4, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.
In this case, the favorable outcomes are the numbers that can be formed using the digits 1, 2, and 3, which are divisible by 4. The total number of possible outcomes is the total number of permutations of the three digits. Therefore, we can calculate the probability. To determine the probability that Jenn's three-digit number is a multiple of 4, we need to consider the divisibility rule for 4. A number is divisible by 4 if the last two digits of the number form a multiple of 4. Given the digits 1, 2, and 3, we can form six different two-digit numbers: 12, 13, 21, 23, 31, and 32. Out of these six numbers, the multiples of 4 are 12 and 32.
Now, let's consider the arrangements of the three digits to form three-digit numbers. Since the first digit cannot be zero, there are six possible arrangements: 123, 132, 213, 231, 312, and 321. Among these arrangements, the numbers 12 and 32 are multiples of 4. Therefore, there are two favorable outcomes.
The total number of possible outcomes is the total number of permutations of the three digits, which is 3! = 6. Thus, the probability that Jenn's three-digit number is a multiple of 4 is 2/6 or 1/3.In conclusion, the probability that Jenn's randomly arranged three-digit number is a multiple of 4 is 1/3.
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What would you call the shape of this formula (C3H8)
An overall shape of a modified tetrahedral with three tetrahedrals embedded within. The three carbons are all central atoms.
Which of the following estimates at a 95% confidence level most likely comes form a small sample?
OptionAStep-by-step explanation:95% confidence interval is obtained using the following formulaCOnfidence interval lower bound = mean - critical value * sigma/sq rt nand upper bound = mean + critical value*sigma/sq rt nThus margin of error = critical value*sigma/rt nvaries indirectly as n provided others are remaining the same.Hence lower sample indicates higher margin of errorOut of 4 options given we find that 1 option has maximum margin of error as 21%Hence option. A is most likely coming from a small sample.
pleas answer correct answer
The equation that can be used to find the value of x is :
\( \boxed{ \boxed{ \frac{x}{7} = \frac{2}{5} }}\)\(x = \dfrac{7 \times 2}{5} \)\(x = \dfrac{14}{5} \)\(x = 2.8\)therefore , correct option is D
Find the diameter of the circle with a circumference of 27 centimeters. Use 3.14 for π. Round to the nearest tenth.
Answer:
Step-by-step explanation:
Circumference= πd = 27 cm
d = 27/π = 8.6 cm
What is the best description of a food chain?
the competition among several species for the same food item
the transfer of energy from one organism to another
Answer:
the transfer of energy from one organism to another
Step-by-step explanation: