A bobcat for 15X30 might work well but you need to protect your lawn from the weight using sheets of plywood. You will need to know where you want to dump that debris, depending on the size of your lot and/or pay for disposal. Do not disturb more soil deeper than 2" or than necessary during the removal of gravel/weeds.
If you want to plant more grass find out what the original grass came from...sod from a local source or seed?
You want as smoothly graded and compacted as possible before seeding or sodding. I recommend sod, easiest, surest and least work. Sod doesn't cost that much more than seeding but worth every cent. If you seed, check out spray seed companies. Less expensive, the company does all of the work after you do the initial grading, raking and rolling. Seeding by owner's hand is a recipe for disappointment. Usually.
Commercial seed spray companies have the best seed mix, this funky colorful mulch to hold the moisture, stabilize all the seedlings and a bit of fertilizer. The best part is that they visit your site before any contract, they teach you how to care for your baby grasses...includes call backs and a warranty. They come back to spray little areas that did not take. You have to keep the soil moist, shallowly moist as the seed germinates. Do not water deeply. A spray seed company will give you the best directions for success or they will come back and spray areas. They will set your automatic irrigation for shallow frequent watering. They really want you to succeed.
After your first mow, we have tons of information to maintain cool season grasses where you should never be adding any other product than balanced fertilizer. Please research care and maintenance of cool season lawns on our site. No other site I know about is credible and unbiased.
One idea is to hire a couple of hardy young men to use shovels and dig this out by hand. 15X30 could be dug out 2" deep only or to the depth the of the soil of the rest of your lawn by two young guys in one hour. 450 sq. ft. of 2" of gravel and debris would be 2.7 yards that needs removed off site, unless you have more than a normal little lot.
Pay them 10 bucks an hour, no more! Grins. Whatever you think you want to pay. Give them one hour to do the job. $20 bucks is nothing. You could motivate them with ice tea, lunch, mentions of bigger better jobs? Depending on their performance? Experience?
Raking and grading and rolling with a water filled roller could get them more time; a half hour should be more than enough. An hour and 1/2 with 2 young people for the job including raking and grading and rolling once more. Then have them lay the sod or have a seed spray company come in and finish the job.
Make sure the soil is at least 4" below the fence. Taking into consideration adding two inches of sod you do not want any soil or mulch or gravel or grass to touch the bottom of your fence.
Check out our other lawn question/answers. Cool season grasses are easy to grow, establish and maintain if you know the simple secrets. Mowing height, how to water, fertilize, aerate and edge. If you understand the basics you will have the most gorgeous, weed free, green, crisp edged lawn on the block, in the entire dang city. Send a picture so that we are not assuming stuff that isn't true?
You could do the removal of 2" gravel, soil and weeds by yourself in 4 hours per day for two days (from my spread sheet for labor costing). If you are less than 50 years old and in shape. Otherwise, double the time. Still have to decide what to do with the debris and you do not want to handle something more than once if at all possible. Dumping debris in a wheel barrow then shoveling that debris out of the wheel barrow into a dump truck is handling the same material twice. Renting a bobcat to use instead of a wheel barrow, fill bucket, bucket lifts up and dumps in trailer. Someone still has to move debris to the front of the trailer. 2 " at 450 sq. ft. makes almost 3 yards. Imagine a washing machine full of soil...that is supposed to be a yard. Make sure you've a truck that can carry the weight of a yard, yard and 1/2? Times 2 runs?
Solid Waste Cleanup Program Weights and Volumes for Project Estimates
Description of Materials Approximate Pounds/Cubic Yard Remarks Burn
Dump Debris/Ash 800-1000 1500-1800 2300 Dry Loose Wet for Dust
Suppression Wet mixed with soil Construction Debris, Asphalt or
Concrete: Loose 2400 Construction Debris, Wood ; Uncompacted 400
Increase up to 100% if compacted using heavy equipment Earth 2100
3000 Loose/Dry. Plus 30% when compacted. Excavated/Wet Gravel or
Crushed Stone Loose/Dry 2600 Increase 20% if wet Household Trash
800 Liquid Waste 1600 202 gal./cubic yard ~ 7 Lbs./Gal. E.g.
Antifreeze, Waste Oil, Solvent Metals, Un-compacted 600 e.g.
Appliances, Metal Siding Sand, Loose/Dry 2400 Increase 20% if damp
and 30% if wet/compacted Stone, Graded 8” max. Loose 2700 e.g.
Gabion Construction. Increase 10% consolidated in place Tire Burn Ash
500-800 Tires, Auto and Pickup 220 Average 10 tires per cubic
yard Tires, OTR See Remarks Average 500 pounds per tire Tires, Truck
480 Average 4 tires per cubic yard Vehicles, Auto and Pickup See
Remarks Use 3000 Pounds/Vehicle Wood Chips, Shredded/Dry Wood
Chips/Bark w/30% Soil 300 800 Yard Waste (Vegetation) Loose 600
Determination of Weights and Volumes of Onsite Materials Volume
Pile volume can best be estimated by determining the area of the base
and then multiplying by the average height of the pile. In many cases
the base of a pile will resemble a rectangle where area is length
times width (L x W). In other cases the pile may more closely resemble
a triangle or other polygon. Use the appropriate geometry to calculate
the base area. For average height, this usually must be estimated
since often it is not prudent to climb a pile to get more exact height
measurements. The height may be estimated by using a known reference
(e.g., fellow inspector) for reference. Cubic yards can be determined
by dividing cubic feet by 27. Depending upon the accuracy of the
assumed measurements, the estimated volume could be within 10-15
percent of the actual volume. Weight
The weight (tonnage) of a pile is determined by multiplying the volume
by the density. CalRecycle’s Solid Waste Cleanup Program has developed
approximate pounds per cubic yard (lbs/cu yd) estimates for various
materials. The actual density depends on the homogeneous nature
(uniformity) of the pile in both void space and material type. Unless
the entire pile can be visualized, it will be difficult to determine
an accurate tonnage estimate. Please note that density values in the
table are general (rough) estimates only and the actual density could
be up to (or exceed) a factor of three (either larger or smaller)
depending upon the actual density of the material. Determination of
maximum weights and volumes that can be received:
Tons permitted to be received per day x 30 days = Maximum amount on
site at any one time Helpful formulas:
_ feet high X feet wide X feet long = _ cubic feet/27 cubic feet per cubic yard = ___ cubic yards
_ cubic yards X 27 cubic feet per cubic yard = _ cubic feet = height X width X length
The pile is 20 feet high X 40 feet wide X 253.1 feet long. This
equates to about 202,479 cubic feet/27 cubic feet per cubic yard =
approximately 7500 cubic yards.
_ cubic yards X _ pounds per cubic yard (waste conversion factor) = ____ pounds/2000 pounds per ton = ____ tons
_ tons X 2000 pounds per ton/pounds per cubic yard = _ cubic yards X 27 cubic feet per cubic yard = height X width X length