Yash designs and builds handmade wooden furniture. He is designing a new box to
have a volume of 48 cubic feet. All the boxes he built are 2 feet wider than their
height and 2 feet longer than their width.
a) Represent the volume of the box with a polynomial in ‘x’.
b) What is the degree of the polynomial so obtained?
c) Identify all the zeroes of the polynomial so obtained.
d) If the height is 2feet find the length and breadth of the box.
e) Write a polynomial in ‘x’ to find the surface area of the box.
a) Represent the volume of the box with a polynomial in ‘x’.
ls it,
x3 -0x2 -4x -48 = 0 .correct sir.
c) Identify all the zeroes of the polynomial so obtained.
sir, how can we find out the zeroes.
If the width is w, then
v = w(w+2)(w-2) = 48
w=4. 4*6*2 = 48
(e) the surface consists of 3 pairs of rectangular faces, so the area is
2(4*6 + 4*2 + 6*2)
seems to me like (d) is useless, since you have already solved the dimensions...
Ausu
a) To represent the volume of the box with a polynomial in 'x', we can start by labeling the height as 'x'.
Given that the boxes Yash builds are 2 feet wider than their height, the width can be expressed as 'x + 2'.
Similarly, the length of the box is described as 2 feet longer than its width, so the length can be represented as 'x + 2 + 2' or 'x + 4'.
The volume of a rectangular box is calculated by multiplying its length, width, and height. In this case, the volume can be expressed as:
Volume = x * (x + 2) * (x + 4)
Therefore, the polynomial representing the volume is:
V(x) = x * (x + 2) * (x + 4)
b) The degree of a polynomial is determined by the highest power of the variable. In this case, the highest power of x in the polynomial V(x) = x * (x + 2) * (x + 4) is 3. Hence, the degree of the polynomial is 3.
c) To identify the zeroes of the polynomial, we set V(x) = 0 and solve for x. However, since the volume cannot be negative in this context, we can ignore the possibility of negative zeroes.
By setting V(x) = 0, we have:
x * (x + 2) * (x + 4) = 0
One or more of the factors (x, x + 2, x + 4) must equal zero to make the entire expression equal zero.
Setting each factor equal to zero, we get:
x = 0 or x + 2 = 0 or x + 4 = 0
From this, the zeroes are x = 0, x = -2 (rejected as it is not a valid solution for the context), and x = -4 (also rejected), given that negative values are not meaningful in this scenario.
So, the valid zero of the polynomial is x = 0.
d) If the height of the box is 2 feet (x = 2), we can find the length and breadth as follows:
To find the length, substitute x = 2 into the equation for length:
Length = x + 4 = 2 + 4 = 6 feet
To find the width, substitute x = 2 into the equation for width:
Width = x + 2 = 2 + 2 = 4 feet
Therefore, if the height is 2 feet, the length of the box is 6 feet, and the width is 4 feet.
e) To write a polynomial in 'x' that represents the surface area of the box, we need to calculate the total area of each face of the rectangular box and sum them up.
The surface area of a rectangular box is given by the formula:
Surface Area = 2*(Length*Width + Length*Height + Width*Height)
Substituting the values of length, width, and height we found earlier:
Surface Area = 2*(6*4 + 6*2 + 4*2)
Simplifying further:
Surface Area = 2*(24 + 12 + 8)
Surface Area = 2*(44)
Surface Area = 88 square feet
Therefore, the polynomial representing the surface area of the box is:
A(x) = 88.