The Great Pumpkin?

Question

The Rhode Island based southern New England Giant Pumpkin Growers club is part of the Great Pumpkin Commonwealth, a kind of world sporting authority for growers. Each year, the club complete at 25 weigh-off in North America and one on Ireland... At the (2005) weigh-off, Rhode island pumpkin took first place , setting a new England record at 1443 pounds, just 26 pounds shy of the world record earlier that year by Pennsylvania grower. A Massachusetts pumpkin came in second at 1333 pounds. Another Rhode Island pumpkin , at 1312 pounds, was heavier then expected, earning third place by 1 ½ pounds.... The Rhode Islander’s top 10 averaged 1174 pounds per pumpkin, topping the charts of the Great Pumpkin Commonwealth... 20 years ago growing a 500 pound pumpkin was considered monumental feat. Now giants regularly up the scale at 1200 to 1400 pounds, 15 feet to 16 feet around, bringing within sight the previously incomprehensible a 1-ton pumpkin.
(1) weights are given for only two of the top ten pumpkins entered from Rhode Island. Write and solve an equation to determine the average weight of the other eight.
(2) Determine the following using the approximate weight of champion pumpkins 20 years ago and the world record weight of the Pennsylvania pumpkin:
a) the average rate of increase in champion weights over the past 20 years
b) the year that we could expect the first 1-ton pumpkin, if growth continues at the average rate
(3) Circumference is a linear measure , while weight is a volumetric, or cubic, measure. Therefore, the weight of an object varies as the cube of its circumference. Estimate the circumference of a 1-ton ( 2000 pound) pumpkin if a 1400 pound pumpkin measures 16 feet around. Assume that pumpkins are spherical and have the same density.
Please, right everything step by step, thanks !

Answer 1

You left something out, but I'll answer anyway. You didn't say champion pumpkins grow in size from year to year. The circumferences could grow linearly or exponentially, or the weights could grow linearly or exponentially. I'll use exponential growth, because (a) there's a big change over 20 years, and (b) with linear growth (and probably exponential growth too), the Pilgrims ate pumpkins the size of peas.

So I'll assume pumpkins grow a certain percent (by weight) each year.

#1. Ten Rhode Island pumpkins weighed 1174 average, and 11,740 total. The top two weighed 2755, so the other eight weighed 8985, or an average of 1123 each.

#2a. Pennsylvania weight is 1469; twenty years ago, champion was 500 lbs.

For linear increase, (1469 - 500)/20 = 48 1/2 lbs/yr.

Better is percentage increase: (log 1469 - log 500)/20 = 0.02340

Inverse log 0.02340 = 1.05536

Pumpkins grow 5.5 percent per year (5.536%)

20-year history of pumpkin weights: 500, 528, 557, 588, 620, 655, 691, 729, 769, 812, 857, 904, 955, 1007, 1063, 1122, 1184, 1250, 1319, 1392, 1469 (count the years!)

and continuing, 1550, 1636, 1727, 1822, 1923, 2030

So the first one-ton pumpkin will be in 2011. To get these numbers, you start with 500 lbs, and multiply each year by 1.055364763 to get the next year's weight.

#2c. x/16 = (2000/1400)^(1/3) = (10/7)^(1/3)

x = 16 (10/7)^(1/3) = 18 feet

Alternate answer ... if pumpkins grow linearly, at 48.45 lbs/yr, then (2000 - 1469)/48.45 = 10.96

so the first one-ton pumpkin will appear 11 years after 2005, or in 2016. But I think the percentage growth model is better than linear growth.

Answer 2

are you asking us to do your homework.
i dont mind to do somebody's homework
but not that long