An open box with a square base (see figure) is to be constructed from 160 square inches of material. The height of the box is 3 inches. What are the dimensions of the box? (Hint: The surface area is S = x2 + 4xh.)
1 . in. (length) × 2 . in. (width) × 3 . in. (height)
S=x²+4xh= x²+12x =160.
x²+12x -160=0
x=- 6±sqrt(36+160) .
x1=-20 (impossible),
x2= +8
1. 8 in;
2. 8 in;
3. 3 in.
Well, let me put on my clown glasses and calculate this for you!
So, we have an open box with a square base and a height of 3 inches. The total surface area is given by the formula S = x^2 + 4xh, where x is the length of the side of the square base and h is the height.
In this case, we know that the surface area is 160 square inches and the height is 3 inches.
So, let's substitute the values into the formula:
160 = x^2 + 4x(3)
Now, we just need to solve this equation. This might require some circus magic!
After doing the calculations (x^2 + 12x - 160 = 0), we find that the dimensions of the box are approximately:
1 inch (length) × 2 inches (width) × 3 inches (height)
So, your open box will be a mini box, perfect for keeping tiny secrets or storing your smallest circus props!
To find the dimensions of the box, we can use the given information about the surface area.
The surface area of an open box with a square base is given by the formula S = x^2 + 4xh, where x is the length of each side of the square base and h is the height of the box.
We are given that the height (h) of the box is 3 inches.
The total amount of material available to construct the box is 160 square inches. This means that the surface area of the box (S) is also 160 square inches.
Plugging in the values into the surface area formula, we have:
160 = x^2 + 4x(3)
Now we need to solve this equation for x.
Expanding the equation, we get:
160 = x^2 + 12x
Rearranging the equation, we have:
x^2 + 12x - 160 = 0
Now we can solve this quadratic equation.
Factoring the equation, we get:
(x - 8)(x + 20) = 0
Setting each factor equal to zero and solving for x, we have:
x - 8 = 0 --> x = 8
x + 20 = 0 --> x = -20
Since the length of a side cannot be negative, we disregard x = -20.
Therefore, the length of each side of the square base (x) is 8 inches.
Since the base is square, the width is also 8 inches.
The height of the box is given as 3 inches.
Therefore, the dimensions of the box are:
Length: 8 inches
Width: 8 inches
Height: 3 inches
To find the dimensions of the box, we need to consider the given information about the surface area and height.
Given that the surface area of the box is S = x^2 + 4xh, where x represents the length and width of the square base, and h represents the height.
In this case, we're given that the height of the box is 3 inches.
Since the box has a square base, the length and width will be equal. Let's represent this value as "a".
The surface area formula can be rewritten as S = a^2 + 4ah.
According to the problem, the total material available for constructing the box is 160 square inches. This means that the surface area (S) is equal to 160.
Substituting these values into the equation, we have:
160 = a^2 + 4ah.
Now, let's solve this equation to find the value of "a" (length and width) which will give us the dimensions of the box.
To solve the equation, we can rearrange it to the form:
a^2 + 4ah - 160 = 0.
This is a quadratic equation, which we can solve by factoring or using the quadratic formula. Let's use factoring.
First, let's try to find two numbers whose product is -160 and whose sum is 4.
By factoring, we find that the numbers are 20 and -8, since 20 * -8 = -160 and 20 + (-8) = 12.
Now, let's rewrite the equation factored:
(a + 20)(a - 8) = 0.
Now we have two possible solutions:
a + 20 = 0 OR a - 8 = 0.
Solving each equation gives us:
a = -20 OR a = 8.
Since measurements cannot be negative, we ignore the negative value, leaving us with the solution:
a = 8.
Therefore, the length and width of the box are both 8 inches, and the height is given as 3 inches.
So, the dimensions of the box are:
Length: 8 inches
Width: 8 inches
Height: 3 inches.