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The volume of a rectangular prism is the product of its length, width, and height. The length of a rectangle prism is 3 feet more than five times its height. The width is 5 feet less than six times its height. Express the length and width in terms of the height h. The length is The width is (Be sure to Write down the expanded form of the expression for the volume: expand the volume expression) (Express the interval in which the In what range must the height fall? He height must be)

Answer :

Answer:

Length is 5 h +3

Width is 6 h -5

Volume is [tex]30h^{3} -7h^{2} -15h[/tex]

Range of height fall [0.8333, ∞)

Step-by-step explanation:

V= l x w x h

Let height of the prism as 'h'.

So, length is 3 ft more than 5 times h.

l = 5 h+3

Now, width is 5 ft less than 6 times h.

w=6 h -5

Volume = (5 h+3)(6 h -5) h.

Expand the volume expression using FOIL method and distribution.

(5 h+3)(6 h-5)= [tex]30h^{2} -25h+18h-15[/tex]

Simplify and then multiply by h.

[tex](30h^{2} -7h-15)h[/tex]

Volume =  [tex]30h^{3} -7h^{2} -15h[/tex]

IF we see the constraint on 'h'

h >0

or

5 h +3 >0

Solve for 'h'

5h > -3

h>[tex]\frac{-3}{5}[/tex]

Negative does not work.

or, 6 h -5 >0

Solve it for h.

6 h >5

h>[tex]\frac{5}{6}[/tex] or h>0.833

So, range of h is greater than 0.833.

In interval notation [0.833, infinity)

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