the The Solution:
Step 1:
We shall split the triangular pyramid into its component surfaces.
3 triangles + 1 triangle.
Step 2:
We shall find the area of the triangle in fig 1., by using Heron's formula.
[tex]\begin{gathered} A=\sqrt[]{s(s-a)(s-b)(s-c)} \\ \text{Where A= area} \\ a=10\text{ ; b= 10 ; c=10} \\ s=\frac{a+b+c}{2}=\frac{10+10+10}{2}=\frac{30}{2}=15 \\ So, \\ A=\sqrt[]{15(15-10)(15-10)(15-10)} \\ A=\sqrt[]{15\times5\times5\times5} \\ A=25\sqrt[]{3}cm^2 \end{gathered}[/tex]Step 3:
We shall state the formula for calculating the total surface area of the triangular pyramid.
[tex]\begin{gathered} \text{SA = Area of fig1 triangle + 3(}\frac{1}{2}bh) \\ \text{Where SA =surface area =214.5 cm}^2 \\ \text{Area of fig 1 triangle = 25}\sqrt[]{3}cm^2 \\ b\text{ =base =10 cm} \\ h=\text{slant height =?} \end{gathered}[/tex]Step 4:
We shall substitute the above values into the formula.
[tex]\begin{gathered} 214.5=25\text{ }\sqrt[]{3}\text{ + 3(}\frac{1}{2}\times10\times h) \\ 214.5=25\text{ }\sqrt[]{3}+3(5h) \\ 214.5=43.3+15h \\ \text{Collecting the like terms, we get} \end{gathered}[/tex][tex]\begin{gathered} 214.5-43.3=15h \\ \text{Dividing both sides by 15, we get} \\ 171.2=15h \\ h=\frac{171.2}{15} \\ h=11.413\approx11.4\text{ cm} \end{gathered}[/tex]Step 4:
The presentation of the Answer.
The slant height is 11.4 centimeters.
Thus, the correct answer is 11.4 centimeters.
