1.a. HG has length 26 m (same as the base of the cuboid), so the length of GM is 13 m. If OG = 30 m, then by the Pythagorean theorem, we have
OM² + (13 m)² = (30 m)² ===> OM = √(30² - 13²) m = √731 m ≈ 27.037 m
1.b. Let P be the center of the square EFGH. Then using the Pythagorean theorem again, we have
OP² + (13 m)² = (√731 m)² ===> OP = √(731 - 13²) m = √562 m
Then the total height of the tower is 70 m + √562 m ≈ 93.707 m
1.c. OM makes an angle θ with the plane EFGH such that
tan(θ) = OP/PM = (√562 m)/(13 m) ===> θ ≈ 61.261°
1.d. The given rate for cleaning doesn't quite make sense, it probably is meant to say the service charges $78 per <em>square</em> meter of the outside of the building. Compute the area of each exposed face (that is, excluding the base):
• Each rectangular face has length 26 m and height 70 m, hence an area of
(26 m) (70 m) = 1820 m²
• Each triangular face has height OM = √731 m and base 26 m, hence an area of
1/2 (√731 m) (26 m) = 13√731 m² ≈ 351.481 m²
Then the total area to be cleaned would be (1820 + 13√731) m², which could cost
($78/m²) ((1820 + 13√731) m²) = $(141,960 + 1014√731) ≈ $169,376.53
2.a. The volume of a sphere with radius <em>r</em> is 4/3 <em>πr</em> ³, so the volume of a hemisphere with the same radius is 2/3 <em>πr</em> ³. If the radius is 3 cm, then the volume of this hemisphere is
2/3 <em>π</em> (3 cm)³ = 2/3 (27<em>π</em> cm³) = 18<em>π</em> cm³
2.b. The volume of a right cone with base radius <em>r</em> and height <em>h</em> is 1/3 <em>πr </em>²<em>h</em>. Note that the cone has the same radius as the hemisphere, <em>r</em> = 3 cm. If this volume is 2/3 as large as the hemisphere's, then solve for the height <em>h</em> :
1/3 <em>π</em> (3 cm)²<em>h</em> = 2/3 (18<em>π</em> cm³)
(3<em>π</em> cm²) <em>h</em> = 12<em>π</em> cm³
<em>h</em> = (12<em>π</em> cm³) / (3<em>π</em> cm²)
<em>h</em> = 4 cm
2.c. Using the Pythagorean theorem,
<em>h</em> ² + <em>r</em> ² = <em>ℓ</em> ² ===> <em>ℓ</em> = √((4 cm)² + (3 cm)²) = √25 cm = 5 cm
2.d. The angle θ between the slant height and the flat of the hemisphere is such that
tan(θ) = <em>r</em>/<em>h</em> = (3 cm)/(4 cm) ===> θ ≈ 36.87°