Question

an image of a castle is shown. the measurements given represent the stone part of the castle. the height of the tower on the right is 80 feet. the diameter of the tower is 20 feet. the building attached to the tower has a length of 200 feet, width of 40 feet, and height of 60 feet. first, determine which shapes best model these two - sections of the castle. then, determine the approximate surface area of the stone part of the castle given the provided dimensions. hint: only the lateral surface area needs to be calculated for the tower. for the building, only calculate the area of three of the faces, which represent the stone parts, and subtract out the area where the building attaches to the tower.

Explanation:

Step1: Identify shapes

The tower is a cylinder with height $h = 80$ feet and diameter $d=20$ feet (radius $r = 10$ feet). The attached - building is a rectangular - prism with length $l = 200$ feet, width $w = 40$ feet, and height $h = 60$ feet.

Step2: Calculate lateral surface area of the tower

The formula for the lateral surface area of a cylinder is $A_{tower}=2\pi rh$. Substituting $r = 10$ and $h = 80$, we get $A_{tower}=2\pi\times10\times80=1600\pi\approx1600\times 3.14 = 5024$ square feet.

Step3: Calculate surface area of the building

We need to calculate the area of three faces of the rectangular - prism and subtract the area where it attaches to the tower. The three faces are: one face with area $A_1=l\times h=200\times60 = 12000$ square feet, another face with area $A_2 = w\times h=40\times60 = 2400$ square feet, and the third face with area $A_3=l\times w=200\times40 = 8000$ square feet. The area where it attaches to the tower is a circle with area $A_{circle}=\pi r^{2}=\pi\times10^{2}=100\pi\approx314$ square feet. The total area of the building's relevant faces is $A_{building}=12000 + 2400+8000-314=22086$ square feet.

Step4: Calculate total surface area

The total surface area of the stone - part of the castle is $A = A_{tower}+A_{building}=5024 + 22086=27110$ square feet. The closest answer among the options is $26400$ square feet.

Answer:

$26,400\ ft^{2}$