You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area
_____________________________________
Among all pairs of numbers whose sum is 20, which pair of numbers will give the maximum product?
Answer
(18,2)
(12,8)
(9,11)
(10,10)
__________________________________
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
f(x) = 11x^3 -6x^2 + x + 3
__________________________________
An object's weight on the moon, M, varies directly as its weight on Earth, E. Neil Armstrong, the first person to step on the moon on July 20, 1969, weighed 360 pounds on Earth (with all of his equipment on) and 60 pounds on the moon. What is the moon weight of a person who weighs 186 pounds on Earth?
_______________________________
The functions is a polynomial function- (true or false).
#4) g(x) = 6x^7 + (pi)x^5 + (2/3)x
___________________________________
The function is a polynomial function. (True or False)
#8) f(x) = x^(1/3) - 4x^2 + 7
To find the length and width of the plot that will maximize the area, we can use the formula for the perimeter of a rectangle: 2(length + width). Since one side of the rectangle is the river, the total length of the fence is 200 feet minus the length of the river.
Let's say the width of the plot is x feet. Then the length of the plot is (200 - x)/2 feet.
The area of a rectangle is given by the formula length x width. So, the area of the plot is (x/2) * ((200 - x)/2).
To find the maximum area, we can take the derivative of the area function with respect to x and set it equal to zero.
Let's differentiate the area function:
dA/dx = [(200 - x)/2] * (1/2) - (x/2) * (1/2)
= (200 - x)/4 - x/4
= (200 - x - x)/4
= (200 - 2x)/4
Setting dA/dx equal to zero:
(200 - 2x)/4 = 0
200 - 2x = 0
2x = 200
x = 100
So, the width of the plot that will maximize the area is 100 feet. Since the length is (200 - x)/2, the length of the plot is (200 - 100)/2 = 100/2 = 50 feet.
Therefore, the length and width of the plot that will maximize the area are 50 feet and 100 feet, respectively.
--------------------------------------------------
Among all pairs of numbers whose sum is 20, the pair of numbers that will give the maximum product is (10, 10).
--------------------------------------------------
To determine the end behavior of the graph of the polynomial function f(x) = 11x^3 - 6x^2 + x + 3, we look at the leading term, which is 11x^3.
Since the leading coefficient is positive (11 > 0) and the degree of the polynomial is odd (3 is odd), the end behavior of the graph is as follows:
- As x approaches negative infinity, f(x) approaches negative infinity.
- As x approaches positive infinity, f(x) approaches positive infinity.
--------------------------------------------------
Since the weight on the moon, M, varies directly as the weight on Earth, E, we can set up a proportion:
M/E = 60/360
To find the moon weight (M) of a person who weighs 186 pounds on Earth (E), we can solve for M:
M/186 = 60/360
Cross multiplying, we get:
M * 360 = 186 * 60
Dividing both sides by 360, we get:
M = (186 * 60) / 360
Simplifying, we have:
M = 31 pounds
Therefore, the moon weight of a person who weighs 186 pounds on Earth is 31 pounds.
--------------------------------------------------
The function g(x) = 6x^7 + (pi)x^5 + (2/3)x is a polynomial function.
--------------------------------------------------
The function f(x) = x^(1/3) - 4x^2 + 7 is not a polynomial function because it contains a fractional exponent (1/3).
To find the length and width of the rectangular plot that will maximize the area, you can use calculus. Let's call the length of the plot 'L' and the width 'W'. Since we know the perimeter is 200 feet, we have the equation 2L + W = 200.
Now, we need to express the area of the rectangle in terms of one variable. The area of a rectangle is given by A = L * W. Using the equation for the perimeter, we can rewrite W as W = 200 - 2L.
Substituting this back into the area equation, we have A = L * (200 - 2L). Expanding this equation, we get A = 200L - 2L^2.
To find the maximum area, we can differentiate the area equation with respect to L and set it equal to zero. The derivative of A with respect to L is dA/dL = 200 - 4L.
Setting dA/dL equal to zero, we have 200 - 4L = 0. Solving for L, we find L = 50.
Now that we know the value of L, we can substitute it back into the equation for the perimeter to find W. Plugging L = 50 into the equation 2L + W = 200, we get 100 + W = 200, which gives us W = 100.
Therefore, the length of the plot that will maximize the area is 50 feet, and the width is 100 feet.
To find the pair of numbers whose sum is 20 and gives the maximum product, you can use algebra.
Let's call the two numbers x and y. We know that x + y = 20.
To find the maximum product, we can express the product of x and y in terms of one variable. The product is given by P = xy.
Using the equation for the sum, we can rewrite y as y = 20 - x.
Substituting this back into the product equation, we have P = x(20 - x). Expanding this equation, we get P = 20x - x^2.
To find the maximum product, we can differentiate the product equation with respect to x and set it equal to zero. The derivative of P with respect to x is dP/dx = 20 - 2x.
Setting dP/dx equal to zero, we have 20 - 2x = 0. Solving for x, we find x = 10.
Now that we know the value of x, we can substitute it back into the equation for the sum to find y. Plugging x = 10 into the equation x + y = 20, we get 10 + y = 20, which gives us y = 10.
Therefore, the pair of numbers that sum up to 20 and gives the maximum product is (10, 10).
To determine the end behavior of the graph of the polynomial function f(x) = 11x^3 - 6x^2 + x + 3, we need to look at the leading term, which is 11x^3.
If the leading term has an odd degree (in this case, x^3), then the graph will have opposite end behaviors. This means that as x approaches negative infinity, the graph will approach negative infinity, and as x approaches positive infinity, the graph will approach positive infinity.
Since the leading term has a positive coefficient (11), the graph will be "upward" in the end behavior.
To find the moon weight of a person who weighs 186 pounds on Earth, we can use the concept of direct variation.
We know that the weight on the moon, M, varies directly as the weight on Earth, E. This means we can set up a proportion: M/E = 60/360.
To find M, the moon weight, when E is 186 pounds, we can cross-multiply and solve for M. This gives us:
M = (60/360) * 186
M = (1/6) * 186
M = 31 pounds
Therefore, the moon weight of a person who weighs 186 pounds on Earth is 31 pounds.
A polynomial function is a function that can be expressed in the form of f(x) = an * x^n + an-1 * x^(n-1) + ... + a1 * x + a0, where n is a non-negative integer and the a's are constants.
Looking at the given function g(x) = 6x^7 + πx^5 + (2/3)x, we can see that it is a polynomial function. The highest power of x in the function is 7, which is a non-negative integer, and all the terms have non-negative integer exponents. Therefore, g(x) is a polynomial function.
Looking at the given function f(x) = x^(1/3) - 4x^2 + 7, we can see that it is not a polynomial function. This is because the term x^(1/3) has a fractional exponent, which is not allowed in a polynomial function. Therefore, f(x) is not a polynomial function.