SOLUTION: A family needs a tent that measures at least 10m^3 and the tent (isosceles triangular prism) they have has a height of 2m, and a base length of 3m. However, the length is unknown,

Question 255687: A family needs a tent that measures at least 10m^3 and the tent (isosceles triangular prism) they have has a height of 2m, and a base length of 3m. However, the length is unknown, but the surface area outside the tent (including bottom) is 38m^2. Will the tent be big enough for them?

Thank you for your help...
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
A family needs a tent that measures at least 10m^3 and the tent (isosceles triangular prism) they have has a height of 2m, and a base length of 3m. However, the length is unknown, but the surface area outside the tent (including bottom) is 38m^2. Will the tent be big enough for them?
What is the volume of a tent with measurements:
h=2m
b=3m
l=?
SA (outside of tent)=38m^2
2 sides for the top, 1 side for bottom, 1 front, 1 back
that is 5 sides
front=back=(1/2)(3)(2)=(1/2)(6)=3 --> total SA front+back 6m^2
((1/2)b)^2+h^2=(width of one of the top sides)^2 let's call it slope^2
((1/2)(3))^2+2^2
1.5^2+4
2.25+4
6.25=slope^2
2.5=slope
top=2*2.5*l=5*l
bottom=3*l
5*l+3*l+6=38
8*l=32
l=4
V=hbl
v=2*3*4=6*4=24 m^3
oh drat, I realized I made an error
volume of the prism standing up right would be area of base times the height
in this case area of one of the triangular sides times the length
V should be 1/2 * base * height * length
V= 1/2 * 3 * 2 * 4
V= 1/2 * 6 * 4
V= 1/2 * 24
V=12 m^3 NOT 24 m^3
family needed a tent that measures at least 10m^3
the tent still satisfies the requirements by 2 m^3

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