The volume of a rectangular is 750 cubic units. The width is 7 units more than the height and length is 1 unit more than eight times the height. Find the dimensions of the solid.
Answers
Answer:
x= height of the rectangular solid, in linear units.
"The width is 7 units more than the height" translates as
x%2B7= width of the rectangular solid, in linear units.
"The length is 1 unit more than eight times the height" translates as
8x%2B1= length of the rectangular solid, in linear units.
Since the volume is the product height%2Awidth%2Alength and equals 750 cubic units,
x%28x%2B7%29%288x%2B1%29=750 is our equation.
It is cubic equation, and we could multiply and look for solutions whichever was we can.
However, should x be an integer, x , %28x%2B7%29 and %288x%2B1%29 would be factors of 750 ,
with x being the smallest of the three, and x%2B7 being 7 units more than x.
750=10%2A75=%282%2A5%29%2A%283%2A25%29=2%2A3%2A5%5E3 .
Since the exponents of the prime factors are 1 , 1 , and 3 ,
750 has %281%2B1%29%2A%281%2B1%29%2A%283%2B1%29=2%2A2%2A4=16 factors.
The smallest of them, in increasing order, are:
1, 2, 3, 5, 6, 10, 15, 25, and 30.
The only pair differing by 7 is 3 and 10 .
If x=3 , x%2B7=3%2B7=10 .
750=10%2A75=10%2A3%2A25=3%2A10%2A25=x%2A%28x%2B7%29%2A%288x%2B1%29
Could it be that system%28x=3%2Cx%2B7=10%2C8x%2B1=25%29 ?
If x=3 , 8x%2B1=8%2A3%2B1=24%2B1=25 , so highlight%28system%28height=3%2Cwidth=10%2Clength=25%29%29 (in linear units, of course).
THE CUBIC EQUATION:
Maybe you were expected to multiply and solve the resulting cubic equation.
That is a long and cumbersome process:
x%28x%2B7%29%288x%2B1%29=750-->x%288x%5E2%2Bx%2B56x%2B7%29=750-->x%288x%5E2%2B57x%2B7%29=750-->8x%5E3%2B57x%5E2%2B7x=750-->8x%5E3%2B57x%5E2%2B7x-750=0
The usual way to solve 8x%5E3%2B57x%5E2%2B7x-750=0 would be to find a rational solution, p%2Fq ,
where p is a factor of 750 and q is a factor of 8 .
Luckily, 3 with system+%7B%7B%7BP=3%2Cq=1 is a rational solution.
That means that the polynomial 8x%5E3%2B57x%5E2%2B7x-750 is divisible by %28x-3%29 .
Dividing, we find
%288x%5E3%2B57x%5E2%2B7x-750%29%2F%28x-3%29=8x%5E2%2B81x%2B250%29<-->8x%5E3%2B57x%5E2%2B7x-750=%28x-3%29%2A8x%5E2%2B81x%2B250%29 .
So the solutions for 8x%5E3%2B57x%5E2%2B7x-750=0 are the solution for x-3=0<-->x=3 ,
plus the solutions to 8x%5E2%2B81x%2B250=0 , if any.
Since 8x%5E2%2B81x%2B250=0 has no solutions, the only solution is highlight%28x=3%29 .
Step-by-step explanation:
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Given:
The volume of the cuboid = 750 cubic units
The width is 7 units more than the height.
The length is 1 unit more than eight times the height.
To Find:
The dimensions of the cuboid
Let us assume:
- Let the length of the cuboid be l unit
- Let the width of the cuboid be b unit
- Let the height of the cuboid be h units
Solution:
- The volume of a cuboid is given as:
"V = lbh"
where l is the length of the cuboid
b is the width of the cuboid
and h is the height of the cuboid
- To solve this question, we will form equations as per the following steps:
1. Relate the dimensions of the cuboid from the given criteria.
b = h + 7 (1)
l = 8h + 1 (2)
2. Substitute the values in the formula for volume.
- From (1) and (2) we will convert the volume into terms of height only.
V = lbh
⇒V = (8h + 1)(h + 7)(h)
⇒ V = (8h² + h)(h + 7)
⇒ V = 8h³ + 56h² + h² + 7h
⇒ V = 8h³ + 57h² + 7h
3. Equate the expression for Volume with the given value.
- It is given that the volume of the cuboid is 750 units.
∴ 8h³ + 57h² + 7h = 750
- Since the given polynomial has the power of 3, we need to use the hit-and-trial method to find the value of h.
- From this method, we find h=3 satisfies our equation:
⇒ 8(3)³ + 57(3)² + 7(3)
⇒ 8×27 + 57×9 + 21
⇒750
∴ LHS = RHS
Therefore, we can conclude that height is 3 units.
From (1), b = h + 7 = 10 units
From (2) l = 8h + 1 = 25 units
Hence, the dimensions of the solid are: