The same sequence may be defined in three ways: 1) f(x) = 3x + 4 for x = {1, 2, 3, }, 2) a1 = 4 and an + 1 = an + 3 for n = {1, 2, 3, } and 3) an = 4 + 3(n − 1) for n = {1, 2, 3, }. they are all supposed to generate the same values, but one version may contain an error. if one version contains an error, identify the one with the error and fix it. a) none of these have errors, they all generate the same sequence. b) f(x) = 3x + 4 for x = {1, 2, 3, } it should say f(x) = 3x + 1 for x = {1, 2, 3, }. c) an = 4 + 3(n − 1) for n = {1, 2, 3, } it should say an = 3 + 4(n − 1) for n = {1, 2, 3, }. d) a1 = 4 and an + 1 = an + 3 for n = {1, 2, 3, } it should say a1 = 3 and an + 1 = an + 4 for n = {1, 2, 3, }.

Find the volume of the solid obtained by rotating the region enclosed by the curves y=1x and y=7^-x about the x-axis. (use symbolic notation and fractions where needed.) volume =

For the feasibility region shown below find the maximum value of the function p=3x+2y

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?