The total surface area of an isosceles triangular prism is 480cm. The base of triangle is 12cm, the overall length is 18cm and the height of the triangle is 7cm. Find the slant length.

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If the base b of the isosceles triangle and its height h are known, the length s of the equal sides is given by s²=h²+b²/4. But s is also the slant length of the prism. If b=12cm and h=7cm as stated in the question, s=√(49+36)=√85=9.22 cm approx. Note this length is greater than either half the base or the height because it’s the hypotenuse of a right triangle formed by the perpendicular bisector of the base.

If A is the total surface area of the prism, A=bh+bd+2sd where d=length of the prism. bh is the total area of the triangular ends, bd the area of the rectangular base, 2sd the total area of the other two rectangular sides. From this formula, s=(A-b(h+d))/2d=(480-12(7+18))/36=180/36=5cm.

But 5cm<√85, so there is an inconstancy in the data or I have misinterpreted the question.

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