Herbert has three lawn-mowing jobs. One lawn is shaped like a trapezoid, one like a triangle, and one like a rectangle. The trapezoidal lawn has bases that are 90 feet and 126 feet and a height of 75 feet. The triangular lawn has a height of 80 feet and a base of 105 feet. The rectangular lawn has a length of 120 feet and width of 90 feet. If Herbert can cut a 160 ft × 70 ft lawn in 2 hours, how long will it take him to mow all three lawns?

4.125 hours

Step-by-step explanation:

Trapezoid shaped lawn :

Bases : 90 ; 126 ; height = 90

Area of trapezium = 1/2(a+b) h ; a and B are the bases and h = height

Area of trapezium = 1/2(90 + 126) * 75

Area = 108 * 90 = 8100 feets

Triangular lawn :

Height = 80 feets ; Base = 105 feets

Area of triangle = 1/2 base * height

Area = 1/2 * 80 * 105 = 4200 feets

Rectangular lawn :

Length = 120 feets ; width = 90 feets

Area of rectangle = Length * width

Area = 120 * 90 = 10800 feet²

Area of all 3 lawns = (8100 + 4200 + 10800) = 23100

In 2 hours ; Herbert can now (160 * 70) = 11200 feets

23100 / 11200 = 2.0625

Tine required to mow all three lawns :

2 hours * 2.0625 = 4.125 hours

= 4 hours, 7 minutes, 30 seconds

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.