Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. if the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope?
(a) 82.
(b) 118.
(c) 120.
(d) 134.
(e) 152.

D = 134

Step-by-step explanation:

The parameters given are  

Mean = 68cm

Median = 84cm

Assuming the length of the ropes starting from the shortest to the longest are  a, b , c , 84, e, f, g

since the length of the shortest rope = a, the length of the longest rope from the question will be = 14 + 4a . therefore g now becomes 14 + 4a

Mean = (total sum of the rope length / number of ropes)

Therefore the total sum of the rope length = mean x number of ropes

= 84 x 7 = 476.

since we are to find the maximum possible length of the longest piece of rope, it is safe to assume that the ropes b and c are of the same length as the first rope since they would still be less than the length of the median rope which is 84cm in length. It is also safe to assume that ropes e and f are of the same length as the median rope since they would still be less than the length of the longest rope and we would still be having 84cm as the value of the median rope length.

With the assumptions above, the seven ropes would now have the lengths of  the ropes as a, a, a, 84, 84, 84,(14 +4a)

 summing up the values above we get 476cm which is the total length as calculated above

a + a + a +84 + 84 + 84 + 14 + 4a = 476

7a + 266 = 476

7a = 476 - 266

7a = 224

a = 30

since a = 30, we can substitute it into the equation of the longest rope (4a +14)

longest rope = (4x 30) + 14 = 134cm

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step-by-step explanation:

i wish i could u

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.