At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete block with a volume of 945 cubic meters. The block's dimensions are x meters high by 12x - 15 meters long by 12x - 21 meters wide. What is the height of the block?
I have no idea how to do this. Can someone please help me?
remember , volume = lwh
x(12x-15)(12x-21) = 945
simplifying a bit
x(3)(4x - 5)(3)(4x-7) = 945
x(4x-5)(4x-7) = 105
16x^3 -48x^2 +35x - 105=0
by grouping ...
16x^2(x - 3) + 35(x-3) = 0
(x-3)(16x^2 + 35) = 0
x = 3 or x^2 = -35/16 , which is not real
so the only real solution is x = 3
so the block is 3 m high,
How do you get 105?
I divided both sides by 9
945÷9=105
notice the (3)(3) is gone on the left side
To find the height of the concrete block, we can set up an equation using the given volume and dimensions.
The volume of the block is given as 945 cubic meters.
The dimensions of the block are given as x meters high, (12x - 15) meters long, and (12x - 21) meters wide.
We can set up the equation as follows:
Volume = Height * Length * Width
945 = x * (12x - 15) * (12x - 21)
To simplify the equation, we can multiply the factors:
945 = x * (144x^2 - 324x + 105)
Now, we need to expand the brackets:
945 = 144x^3 - 324x^2 + 105x
Next, we can rearrange the equation to isolate the x term:
144x^3 - 324x^2 + 105x - 945 = 0
To solve this equation, we can use various methods such as factoring, synthetic division, or numerical methods like Newton's method. However, this equation is quite complex and does not have a simple factorization. Therefore, we will use a numerical method to find an approximate solution.
You can use computational tools like a graphing calculator or software programs to find the value of x that satisfies the equation. Alternatively, you can use online equation solvers or consult with a math teacher or tutor for assistance.
Once you have found the value of x, you can substitute it back into the equation x to calculate the height of the block.