The length of a box is 2 centimeters less than its height. the width of the box is 7 centimeters more than its height. If the box had has a volume of 180 cubic centimeters, what is its surface area?

Question

Evelyn Arboleda · Accepted Answer

Let the height of the box be #h# cm
Then its Length will be #(h-2)# cm and its width will be #(h+7)# cm
So by the condtion of the problem  #(h-2)xx(h+7)xxh=180# #=>(h^2-2h)xx(h+7)=180# #=>h^3-2h^2+7h^2-14h-180=0# #=>h^3+5h^2-14h-180=0# For #h=5# LHS becomes zero 
Hence #(h-5)# is factor of LHS So 
#h^3-5h^2+10h^2-50h+36h-180=0#
#=>h^2(h-5)+10h(h-5)+36(h-5)=0# #=>(h-5)(h^2+10h+36)=0# So Height #h=5# cm Now  Length #=(5-2)=3# cm Width #=5+7=12# cm So the surface area becomes #2(3xx12+12xx5+3xx5)=222cm^2#

Carson Arnold · Answer

undefined Let's denote the height of the box as $h$ centimeters.

Given:
1. The length of the box is $2$ centimeters less than its height, so the length $l$ can be expressed as $h - 2$ centimeters.
2. The width of the box is $7$ centimeters more than its height, so the width $w$ can be expressed as $h + 7$ centimeters.
3. The volume of the box is $180$ cubic centimeters, so we have the equation $l \times w \times h = 180$.

Substituting the expressions for $l$ and $w$ into the volume equation, we get:
$$ (h - 2) \times (h + 7) \times h = 180 $$

Expanding and simplifying:
$$ (h^2 + 7h - 2h - 14) \times h = 180 $$
$$ (h^2 + 5h - 14) \times h = 180 $$
$$ h^3 + 5h^2 - 14h - 180 = 0 $$

Now, we need to find the value of $h$ that satisfies this equation. By trying different values or using numerical methods, we find that $h = 6$ is a solution to this equation.

So, the height of the box is $6$ centimeters.

Now, we can find the length and width:
$$ l = h - 2 = 6 - 2 = 4 \text{ centimeters} $$
$$ w = h + 7 = 6 + 7 = 13 \text{ centimeters} $$

Now that we have the dimensions of the box, we can find its surface area. The surface area $A$ of a rectangular box is given by the formula:
$$ A = 2lw + 2lh + 2wh $$

Substituting the values we found:
$$ A = 2(4 \times 13) + 2(4 \times 6) + 2(13 \times 6) $$
$$ A = 2(52) + 2(24) + 2(78) $$
$$ A = 104 + 48 + 156 $$
$$ A = 308 \text{ square centimeters} $$

Therefore, the surface area of the box is $308$ square centimeters.