An open box is to be made from a flat piece of material 18 inches long and 2 inches wide by cutting equal squares of length xfrom the corners and folding up the sides.
Write the volume Vof the box as a function of x. Leave it as a product of factors, do not multiply out the factors.
V=
If we write the domain of the box as an open interval in the form (a,b), then what is a=?
a=
and what is b=?
I got V=x(18-2x)(2-2x)
and a=0
b=9
but it says that its not correct
the equation you got is correct,, =)
first, note that x (which is the height of the box) cannot be negative, since there is no negative height,,
equating V to zero,
x=0, 1, 9
now, get some values in between (and if you want, get also a value greater than 9), and check if the dimensions will all be positive, and V will be positive as well,,
example,,
if x = 0.5, V = (0.5)(17)(1) = 8.5
if x = 4, V = (4)(10)(-6) = -240
if x = 10, V = (10)(-2)(-18) = 360
note that even if the volume that we got for x = 10 is positive, the length and width are negative, which cannot happen,,
therefore, domain is (0,1)
so there,, i hope i was able to help.. =)
Let's solve the problem step by step:
1. The volume V of the box can be found by multiplying the length, width, and height of the box. The length and width of the box are determined by the dimensions of the flat piece of material.
2. To find the height of the box, we need to consider that equal squares of length x are being cut from each corner and folded up to form the sides.
3. Since the length of the material is 18 inches and two squares of length x are cut from each end, the length of the box will be (18 - 2x) inches.
4. Similarly, since the width of the material is 2 inches and two squares of length x are cut from each end, the width of the box will be (2 - 2x) inches.
5. The height of the box is simply x inches.
Putting this together, the volume of the box V can be calculated as:
V = length * width * height
V = (18 - 2x) * (2 - 2x) * x
However, we are asked to leave the factors in the factored form without multiplying them out. So the correct answer for the volume V is:
V = x(18 - 2x)(2 - 2x)
Now let's determine the domain of the box, which is the range of valid values for x.
6. Since the squares are being cut from the corners, x cannot exceed half of the length or width of the material.
7. The length of the material is 18 inches, so the maximum value of x is 9 inches (half of 18).
Therefore, the domain of x is (0,9).
So, the correct answers are:
V = x(18 - 2x)(2 - 2x)
a = 0
b = 9
To find the volume of the box, you correctly determined that it can be expressed as V = x(18 - 2x)(2 - 2x). However, the values you provided for a and b are not correct. Let me explain how to find the correct values.
To determine the domain of the box, we need to consider the length and width of the initial flat piece of material. In this case, the length of the material is 18 inches, and the width is 2 inches.
When we cut equal squares of length x from each corner and fold up the sides, the resulting length of the box will be (18 - 2x) inches, and the width will be (2 - 2x) inches.
Since the length and width of a box cannot be negative, we need to ensure that (18 - 2x) and (2 - 2x) are positive. This means we should set the inequalities as follows:
18 - 2x > 0 (for the length)
2 - 2x > 0 (for the width)
To solve the first inequality:
18 - 2x > 0
2x < 18
x < 9
To solve the second inequality:
2 - 2x > 0
-x > -1
x < 1
Now, we have found that x must be less than 9 and less than 1. The largest value that satisfies both inequalities is x = 1.
Therefore, the correct values for a and b are: a = 0 (the lower bound of the domain), and b = 1 (the upper bound of the domain).
Thus, the final answer is:
V = x(18 - 2x)(2 - 2x) for 0 < x < 1.
Your equation V=x(18-2x)(2-2x) , which is correct, describes the volume for any given x
clearly V has to be a positive number.
notice that for x=9 and x=1, V is zero.
for all values of x between 1 and 9 , V is positive,
for x<1 and x > 9 , V is negative.
try a number between 1 and 9
Draw your conclusion from my hints.