SOLUTION: I am having problems to solve the following problems 1) A lot is in the shape of a right triangle. The shorter leg measures 120 m. The hypotenuse is 40 m longer than the leng

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Question 86898: I am having problems to solve the following problems
1) A lot is in the shape of a right triangle. The shorter leg measures 120 m. The hypotenuse is 40 m longer than the length of the longer leg. How long is the longer leg?

2) Your company uses the quadratic model y = -4.5x2 + 150x to represent the average number of new customers who will be signed on (x) weeks after the release of your new service. How many new customers can you expect to gain in week 4?

3) The position of an object moving in a straight line is given by s = 2t2 - 3t, where s is in meters and t is the time in seconds the object has been in motion. How long (to the nearest tenth) will it take the object to move 13 meters?
4) The height of a box is 3 inches. Its length is 4 inches more than its width. Find the length if the volume is 180?

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
1)
Because the shape of the lot is a right triangle, you can use the Pythagorean theorem c%5E2+=+a%5E2+%2B+b%5E2 to solve this problem.
Let a = 120 m (this is the length of the shorter leg)
Let b = ? (this is the longer leg whose length we need to find)
Let c = b+40 (this is the length of the hypotenuse)
Making the appropriate substitutions into the Pythagorean theorem formula, we get:
%28b%2B40%29%5E2+=+120%5E2+%2B+b%5E2 Simplify and solve for b.
b%5E2+%2B+80b+%2B+1600+=+14400+%2B+b%5E2 Subtract b%5E2 from both sides.
80b+%2B+1600+=+14400 Subtract 1600 from both sides.
80b+=+12800 Finally, divide both sides by 80.
b+=+160m. This is the length of the longer leg.
The length of the hypotenuse is 160 + 40 = 200 m.
2)
To find the expected number of new customers gained in week 4, you first need to find the number of new customers gained 3 weeks after the release of the new service and subtract this number from the number of new customers gained 4 weeks after the release of the new service. You will then have the number of new customers gained in week 4.
To find the number of new customers (y) signed on 3 weeks after the release, let x = 3.
y+=+-4.5x%5E2%2B150x
y+=+-4.5%283%5E2%29+%2B+150%283%29
y+=+-4.5%289%29+%2B+450
y+=+-40.5+%2B+450
y+=+409.5 average number of new customers signed on after 3 weeks.
Average number of new customer signed on 4 weeks after the release:
y+=+-4.5%284%5E2%29+%2B+150%284%29
y+=+-72+%2B+600
y+=+528 average number of new customers signed on after 4 weeks.
Take the difference of these two to find the number of cutomers gained in week 4.
528 = 409.5 = 118.5 new customers gained in week 4.
3)
To find out how long (t) it will take the object to move 13 meters, substitute s = 13 and solve for t.
s+=+2t%5E2-3t
13+=+2t%5E2-3t Subtract 13 from both sides.
2t%5E2-3t-13+=+0 Solve using the quadratic formulat+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a In this problem, a = 2, b = -3, and c = -13
t+=+%28-%28-3%29%2B-sqrt%28%28-3%29%5E2-4%282%29%28-13%29%29%29%2F2%282%29
t+=+%283%2B-sqrt%289-%28-104%29%29%29%2F4
t+=+%283%2B-sqrt%28113%29%29%2F4
t+=+3.4 or t+=+-1.9 Discard the negative answer because a negative time is not meaningful in this problem.
It will take the object 3.4 seconds to move a distance of 13 meters.
4)
Height = 3"
Width = W"
Length = (W+4)"
Volume = 180 cu.in.
Start with the formula for the volume of a rectangular prism (a box):
V+=+L%2AW%2AH Make the appropriate substitutions from the given data.
180+=+%28W%2B4%29%28W%29%283%29 Simplify and solve for W.
180+=+3W%5E2%2B12W Subtract 180 from both sides.
3W%5E2%2B12W-180+=+0 Divide both sides by 3 to simplify.
W%5E2%2B4W-60+=+0 Factor this quadratic equation.
%28W%2B10%29%28W-6%29+=+0
W+=+-10 or W+=+6 Discard the negative solution as a negative W is not meaningful in this problem.
The width is 6 meters.
The length is W+4 = 6+4 = 10 meters.