You were assigned to construct
qn
open-top box with a square base
from
two materials, one
for
the
bottom and one
for
the sides. The volume of a box is 78 cubic inches. The cost of the material for
the
bottom is Php 4 per square inch, while the cost of the material
for
the sides is Php 3 per square inch.
To construct an open-top box with a square base, you will need to calculate the dimensions of the box, determine the materials needed for the bottom and sides, and calculate the total cost of the materials.
Step 1: Calculate the dimensions of the box.
Since the box has a square base, let's assume the length and width of the base are both denoted as "x". The height of the box can be denoted as "h".
Step 2: Calculate the volume of the box.
The volume of a rectangular prism can be calculated by multiplying the length, width, and height. In this case, the volume is given as 78 cubic inches. So we have:
x * x * h = 78
Step 3: Determine the materials needed for the bottom and sides.
The material for the bottom of the box is calculated by finding the area of the square base. The material for the sides is calculated by finding the area of the four vertical rectangles. So we have:
Bottom material: x * x
Side material: 4 * x * h
Step 4: Calculate the cost of the materials.
The cost of the material for the bottom is Php 4 per square inch, and the cost of the material for the sides is Php 3 per square inch. So we have:
Bottom cost: Php 4 * (x * x)
Side cost: Php 3 * (4 * x * h)
Step 5: Find the value of x and h.
To determine the values of x and h, we need to solve the equation from Step 2.
Step 6: Calculate the total cost of the materials.
To find the total cost, add the cost of the bottom material and the side material. So we have:
Total cost = Bottom cost + Side cost
Please provide the value of x or any other relevant information to proceed further with the calculations.
To construct an open-top box with a square base, we need to determine the dimensions of the box and calculate the cost of the materials.
Let's start by labeling the variables:
- Let x be the length of one side of the square base.
- Let h be the height of the box.
Now, we can calculate the volume of the box using the formula for the volume of a rectangular prism:
Volume = length * width * height
Since the base is square, the length and width of the base are both x. Therefore, the volume formula becomes:
Volume = x * x * h = x^2 * h
According to the given information, the volume of the box is 78 cubic inches. So we can write the equation:
x^2 * h = 78
Next, let's calculate the cost of the materials. We are given that the cost of the material for the bottom (the square base) is Php 4 per square inch and the cost of the material for the sides is Php 3 per square inch.
The cost of the material for the bottom can be calculated by multiplying the area of the square base by the cost per square inch. Since the base is a square with sides of length x, the area of the base is x^2:
Cost of bottom = x^2 * 4 = 4x^2
The cost of the material for the sides can be calculated by multiplying the surface area of the four sides (excluding the bottom) by the cost per square inch. The surface area of the sides is given by adding the area of each side. Since there are four sides, each with a length of x and height of h, the surface area of the sides is 4xh:
Cost of sides = 4xh * 3 = 12xh
Now, let's summarize the constraints:
- Volume = x^2 * h = 78
- Cost = Cost of bottom + Cost of sides = 4x^2 + 12xh
To find the values of x and h that satisfy both constraints, we can use substitution or elimination methods by solving these equations simultaneously.