SAT Question 9fdc4eaa: Area and Volume answer and explanation with Desmos solution

Choice A is correct. The total surface area of a right square pyramid is the sum of the combined surface area of its four lateral faces and the area of its square base. It’s given that the right square pyramid has a total surface area of $70,560$ square inches, and the combined surface area of the four lateral faces of this pyramid is $38,160$ square inches. It follows that the area of the square base is the difference between the total surface area and the combined surface area of the four lateral faces, or $(70,560-38,160)$ square inches, which is equal to $32,400$ square inches. The area, $B$, of the base of a square pyramid is given by the equation $B=s^2$, where $s$ is the side length of the base. Substituting $32,400$ for $B$ in this equation yields $32,400=s^2$. Taking the positive square root of both sides of this equation yields $180=s$. The four lateral faces of a square pyramid are congruent triangles. It follows that the area of each lateral face is $\frac{38,160}{4}$, or $9,540$, square inches. The area, $A$, of each lateral face of a square pyramid is given by the equation $A=\frac{1}{2}sl$, where $s$ is the side length of the pyramid’s base and $l$ is the pyramid’s slant height. Substituting $9,540$ for $A$ and $180$ for $s$ in this equation yields $9,540=\frac{1}{2}(180)l$, or $9,540=90l$. Dividing both sides of this equation by $90$ yields $106=l$. In the right square pyramid, a right triangle can be drawn, where the lengths of the legs correspond to the pyramid's height, $h$, and half the side length of its base, $\frac{1}{2}s$, and the length of the hypotenuse corresponds to the pyramid's slant height, $l$. It follows that the relationship between $h$, $s$, and $l$ is given by the equation $h^2+(\frac{1}{2}s{)}^2=l^2$. Substituting $180$ for $s$ and $106$ for $l$ in this equation yields $h^2+((\frac{1}{2})(180){)}^2=(106{)}^2$, or $h^2+8,100=11,236$. Subtracting $8,100$ from both sides of this equation yields $h^2=3,136$. Taking the positive square root of both sides of the equation yields $h=56$. Therefore, the height, in inches, of this pyramid is $56$.