answer:1. **Let ( r ) be the radius of the sphere.** 2. **Let ( x ) be the radius of the base of the cone formed by the light source and the illuminated cap on the sphere.** 3. **Let ( l ) be the slant height of the cone.** 4. **Let ( h = r - y ) be the height of the spherical cap formed by the intersection of the cone with the sphere.** 5. **Given that the distance from the light source to the sphere is ( 3r ).** 6. **We need to find the values of ( x ) and ( y ) and then relate the surface area of the illuminated part of the sphere (( F )) to the lateral surface area of the cone (( P )).** 7. **We start with the Pythagorean theorem applied to the triangle formed by the radius of the base of the cone ( x ), the segment ( y ) from the sphere center to the base of the cap, and the segment from the sphere center to the tip of the cone:** [ x^2 = r^2 - y^2 ] 8. **The distance from the light source to the tangent point on the sphere is ( 3r - r = 2r ). Therefore, ( x ) and ( y ) should satisfy:** [ r^2 - y^2 = (4r - y)y ] 9. **Simplify the equation:** [ r^2 - y^2 = 4ry - y^2 ] [ r^2 = 4ry ] [ y = frac{r}{4} ] 10. **Next, we need to find ( x ):** [ x^2 = r^2 - left(frac{r}{4}right)^2 ] [ x^2 = r^2 - frac{r^2}{16} ] [ x^2 = frac{16r^2 - r^2}{16} ] [ x^2 = frac{15r^2}{16} ] [ x = frac{r}{4} sqrt{15} ] 11. **Given that the slant height ( l ) of the cone is 4 times the base radius ( x ):** [ l = 4x = 4 cdot frac{r}{4} sqrt{15} = r sqrt{15} ] 12. **Calculate the lateral surface area of the cone ( P ):** [ P = pi x l = pi cdot left(frac{r}{4} sqrt{15}right) cdot r sqrt{15} = pi cdot left(frac{r}{4}right) cdot 15r = frac{15 pi r^2}{4} ] 13. **Calculate the surface area of the illuminated cap ( F ):** [ F = 2 pi r (r - y) = 2 pi r left(r - frac{r}{4}right) = 2 pi r cdot frac{3r}{4} = frac{3 pi r^2}{2} ] 14. **Now, find the ratio ( frac{F}{P} ):** [ frac{F}{P} = frac{frac{3 pi r^2}{2}}{frac{15 pi r^2}{4}} = frac{3}{2} cdot frac{4}{15} = frac{6}{15} = frac{2}{5} ] **Conclusion:** [ boxed{frac{2}{5}} ]

answer:First, find the volume of the original pyramid. The base area (B) of a square pyramid is calculated as the square of the base edge: [ B = 16^2 = 256 text{ cm}^2. ] The formula for the volume (V) of a pyramid is: [ V = frac{1}{3}Bh = frac{1}{3}(256)(10) = frac{2560}{3} text{ cm}^3. ] The smaller pyramid is similar to the original, with a scale factor of (frac{1}{2}). The volume of a shape scaled by a factor (k) changes by (k^3) in three dimensions: [ V_{text{small}} = left(frac{1}{2}right)^3 V = frac{1}{8}V = frac{1}{8}left(frac{2560}{3}right) = frac{2560}{24} text{ cm}^3. ] The volume of the frustum (V_{text{frustum}}) is the volume of the original pyramid minus the volume of the smaller pyramid: [ V_{text{frustum}} = V - V_{text{small}} = frac{2560}{3} - frac{2560}{24} = frac{7680}{24} - frac{2560}{24} = frac{5120}{24} = frac{2560}{12} text{ cm}^3. ] [ V_{text{frustum}} = frac{2560}{12} = 213.33 text{ cm}^3. ] [ boxed{213.33 text{ cubic centimeters}} ]

answer:First, let's calculate the total number of apples Jessica needs for the 3 pies. Since each pie contains 8 servings and each serving requires 1.5 apples, we can calculate the number of apples per pie: 8 servings/pie * 1.5 apples/serving = 12 apples/pie Now, since Jessica is making 3 pies, we multiply the number of apples per pie by the number of pies: 12 apples/pie * 3 pies = 36 apples If all the pies are finished and on average each guest eats 3 apples, we can find out the number of guests by dividing the total number of apples by the average number of apples each guest eats: 36 apples / 3 apples/guest = 12 guests So, Jessica has boxed{12} guests.

answer:To find the total number of corn cobs grown, we need to calculate the number of corn cobs in each field and then add them together. For the first field with 13 full rows of corn cobs: 13 rows * 4 corn cobs per row = 52 corn cobs For the second field with 16 full rows of corn cobs: 16 rows * 4 corn cobs per row = 64 corn cobs Now, we add the number of corn cobs from both fields together: 52 corn cobs + 64 corn cobs = 116 corn cobs Therefore, the farm has grown a total of boxed{116} corn cobs.