the volume of a rectangular solid is given by expressions given below.in each case find the dimentions of the solid.
a)15x^2-51x+18.
b)x^3+13x^2+32x+20.
c)x^2-5x/2-3/2.
d)x^4+x^3-x-1
plz everyone help me out.till 2 o'clock...plz plz plz
Factorize:
a)15x^2-51x+18.
=15x^2-45x -6x+18
=15x(x-3) - 6(x-3)
=(15x-6)(x-3)
b) x^3+13x^2+32x+20.
Notice that by changing the sign of odd powers, the coefficients add up to zero:
-1+13-32+20=0
So x+1 is a factor.
Divide by x+1 gives
x^2+12x+20
which factorizes readily to
(x+2)(x+10)
So the whole factorization is:
(x+1)(x+2)(x+10)
(c) x^2-5x/2-3/2.
=(1/2)(2x^2-5x-3)
=(1/2)[2x^2+x - 3(2x+1)]
=(1/2)[x(2x+1)-3(2x+1)]
=(1/2)(x-3)(2x+1)
(d) will be left as a practice for you.
To find the dimensions of a rectangular solid given the volume expression, we need to factorize the volume expression into three parts representing the dimensions of the solid. Let's solve each case step by step:
a) 15x^2 - 51x + 18
To factorize this expression, we look for two factors of 15x^2 that multiply to give us 15x^2 and also two factors of 18 that multiply to give us 18. Here's how we can factorize it:
15x^2 can be factored as 3x * 5x.
18 can be factored as 3 * 6.
Now we need to distribute these factors across the three dimensions. For a rectangular solid, let's assume the dimensions are length (L), width (W), and height (H). So, we can set up the following equations:
L = 3x
W = 5x
H = 6
Hence, the dimensions of the rectangular solid are L = 3x, W = 5x, and H = 6.
b) x^3 + 13x^2 + 32x + 20
To factorize this expression, we again look for two factors of x^3 that multiply to give us x^3 and two factors of 20 that multiply to give us 20. Here's how we can factorize it:
x^3 can be factored as x * x^2.
20 can be factored as 4 * 5 or 2 * 10 or 1 * 20.
Distributing these factors across the three dimensions, we get:
L = x
W = x^2
H = 4
Hence, the dimensions of the rectangular solid are L = x, W = x^2, and H = 4.
c) x^2 - (5x/2) - (3/2)
To factorize this expression, we look for two factors of x^2 that multiply to give us x^2 and also two factors of -3/2 that multiply to give us -3/2. Here's how we can factorize it:
x^2 can be factored as x * x.
-3/2 can be factored as (-1/2) * 3/2.
Distributing these factors across the three dimensions, we get:
L = x
W = x
H = (-1/2) * 3/2
Simplifying the last equation, we have:
H = -3/4
Hence, the dimensions of the rectangular solid are L = x, W = x, and H = -3/4.
d) x^4 + x^3 - x - 1
To factorize this expression, we look for two factors of x^4 that multiply to give us x^4 and two factors of -1 that multiply to give us -1. Here's how we can factorize it:
x^4 can be factored as x^2 * x^2, or x * x * x * x.
-1 can only be factored as -1 * 1 or 1 * -1.
Distributing these factors across the three dimensions, we get:
L = x^2
W = x^2
H = 1
Hence, the dimensions of the rectangular solid are L = x^2, W = x^2, and H = 1.