Question

find the surface area and volume of solids
3.
find the surface area ______

Explanation:

Step1: Find base area

The base is a square with side length $s = 4$ in. The area of the base $B$ is given by $B=s^{2}$, so $B = 4^{2}=16$ in².

Step2: Find area of one triangular face

The formula for the area of a triangle is $A=\frac{1}{2}bh$. The base of each triangular face $b = 4$ in and the slant - height $l=6.3$ in. So the area of one triangular face $A_{face}=\frac{1}{2}\times4\times6.3 = 12.6$ in².

Step3: Find total surface area

A square - based pyramid has 4 triangular faces and 1 square base. The total surface area $SA$ is $SA = B+4A_{face}$. Substitute $B = 16$ in² and $A_{face}=12.6$ in² into the formula: $SA=16 + 4\times12.6=16+50.4 = 66.4$ in².

Step4: Find the height of the pyramid (to find volume)

We can use the Pythagorean theorem to find the height $h$ of the pyramid. Let the slant - height be $l = 6.3$ in and half of the base side length be $a=\frac{4}{2}=2$ in. Then $h=\sqrt{l^{2}-a^{2}}=\sqrt{6.3^{2}-2^{2}}=\sqrt{39.69 - 4}=\sqrt{35.69}\approx5.97$ in.

Step5: Find the volume of the pyramid

The volume of a pyramid is given by $V=\frac{1}{3}Bh$, where $B = 16$ in² and $h\approx5.97$ in. So $V=\frac{1}{3}\times16\times5.97=\frac{95.52}{3}=31.84$ in³.

Answer:

Surface area: $66.4$ in²
Volume: $31.84$ in³