An open rectangular tank is made of steel plate of area 1440cm it height is twice its width if the depth of the tank is 4 m less than width. Find its length
Let's assign variables to each dimension of the tank:
- Width = x
- Height = 2x
- Depth = x - 4
To find the length, we need to use the formula for the surface area of a rectangular tank:
Surface area = 2(length × width) + 2(length × height) + 2(width × height)
Since one of the sides is open, we won't consider that in the surface area formula. The surface area is given as 1440 cm².
1440 = 2(length × x) + 2(length × 2x) + 2(x × 2x)
1440 = 2x(length) + 4x(length) + 4x²
1440 = 6x(length) + 4x²
We also know that the area of a rectangle is given by length × width. In this case, the area is 1440 cm². Setting up the equation:
1440 = length × x
We can solve this equation for length by substituting length with 1440/x in the surface area equation:
1440 = 6x(1440/x) + 4x²
1440 = 8640/x + 4x²
To simplify, we can clear the fractions by multiplying both sides of the equation by x:
1440x = 8640 + 4x³
Rearranging the equation:
4x³ - 1440x + 8640 = 0
Factoring out a common factor of 4:
4(x³ - 360x + 2160) = 0
Dividing through by 4:
x³ - 360x + 2160 = 0
We can now use numerical or graphical methods to find the value of x that satisfies this equation. However, it is apparent that x = 12 is a solution.
So the width of the tank is 12 cm. The height is twice the width, so it is 2 * 12 = 24 cm. The depth is 4 m less than the width, so it is 12 - 4 = 8 cm.
Finally, we can find the length using the area equation:
1440 = length × 12
length = 1440 / 12 = 120 cm.
Therefore, the length of the tank is 120 cm.
Let's solve the problem step-by-step:
Step 1: Let's assume the width of the tank as "x" (in cm).
Step 2: As given in the problem, the height of the tank is twice its width. Therefore, the height would be 2x cm.
Step 3: The area of the steel plate is given as 1440 cm². This can be calculated by multiplying the length, width, and height:
Area = Length * Width * Height
1440 = Length * x * 2x
Step 4: Simplifying the equation:
1440 = 2x² * Length
Step 5: We are given that the depth of the tank is 4 meters less than the width. Therefore, the depth would be (x - 400) cm.
Step 6: Now, we can substitute the value of the depth and the height (2x cm) into the equation:
1440 = 2x² * Length
1440 = 2x * (x - 400) * Length
Step 7: Simplifying the equation further:
1440 = 2x² * (x - 400)
Step 8: Expand the equation:
1440 = 2x³ - 800x²
Step 9: Rearrange the equation to make it equal to zero:
2x³ - 800x² - 1440 = 0
Step 10: Now, we can solve this cubic equation to find the possible values of x using mathematical methods or calculators.
Note: The solution of the cubic equation will give us the value(s) of x, which represents the width of the tank. The length can be calculated using the formula:
Length = Area / (Width * Height)
Substituting the found values of width and height will give us the length.