An interior designer is buying fabric to cover throw pillows for a master bedroom. she needs material to cover 6 pillows—3 rectangle-shaped, 2 cylinder-shaped, and one sphere-shaped. the rectangles are 18 inches by 12 inches by 2 inches. the cylinders are 12 inches high and have a radius of 7 inches. the sphere has a radius of 10 inches. a. what is the surface area of each rectangle-shaped pillow? b. what is the surface area of each cylinder-shaped pillow? c. what is the surface area of the sphere-shaped pillow? d. what is the total surface area for all 6 pillows? the designer also needs to buy stuffing for the pillows.  e. what is the volume of each rectangle-shaped pillow? f. what is the volume of each cylinder-shaped pillow? g. what is the volume of the sphere-shaped pillow? h. what is the total volume needed to stuff all the pillows

Step-by-step explanation:

a-c. The area formulas for these figures are ...

  rectangular prism: A = 2(lw +h(l+w))

  cylinder: A = 2πr(r +h)

  sphere: A = 4πr^2

d. The total will be the sum of products: area of each pillow times the number of that type

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e-g. The volume formulas for these figures are ...

  rectangular prism: V = lwh

  cylinder: V = πr^2h

  sphere: V = (4π/3)r^3

h. As with area, the total volume is the sum of products: volume of each pillow times the number of that type.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.