Problem Solving?

Question

1.Each plant of a certain variety yields 50 seeds in the early fall and then dies. Only 40 percent of these seeds produce plants the following summer and the remainder never produce plants. At this rate, a single plant yielding seeds in 1986 will produce how many plants as descendants in 1989?

2.A figure has a 10-inch piece of string attached to the ends of a 6-inch rod. If the string is pulled so as to form the sides of a triangle whose base is the rod, then the greatest possible area of such a triangle is?
I asked it already and one person got 11.31 but I would like to know the steps.

Thank you.

Answer 1

1. You would multiply the original number of seeds by 40% three times (1986-89). So it's 50 x 0.40 x 0.40 x 0.40 = 3.2 plants. You would have to say only 3 plants, as you can't have a fraction of a plant.

2. The greatest area for a triangle with a fixed base would be an isoceles triangle, so divide the string length in half to get the length of the two equal sides; that would make them 5 inches each. If you draw a diagram of a 5-5-6 triangle with the 6 inch side as the base, then drop a perpendicular from the apex to the middle of the base, you make two right-angled triangles with 3 as the base and 5 as the hypotenuse.

You should recognize these as 3-4-5 right triangles, with 4 inches being the height. The area of each would be 1/2 x b x h, but you have two of them, so the total area is just 3 x 4 or 12 square inches. That would be the maximum area.

Answer 2

1. 50 * .4^3, three being the number of years of production (1987 to 1989) = 50 * .064 = 3.2, so 3 plants in 1989.

2. 10 inch string would produce two 5 inch sides. The triangle is actually two right triangles, with sides of 3 (half the rod) and the height, with a hypotenuse of 5. So the height is 4 (3-4-5 triangle), and the area is 6 * 4/2 = 12 sq. inches.

Answer 3

1. This appears to be a trick question. A single plant that produces seeds in 1986 will die and produce no plants as descendants in 1989.
2. Create an isoscelese triangle with the rod and the string. The height of the triangle will be 4 inches. If you draw the height as a line, this breaks the original triangle into 2 identical 3-4-5 trianges each with an area of 6 sq inches (12 sq inches total).

Answer 4

In 1987 there would be 20 new plants
In 1998 there would be 400 plants.
In 1989 there would 8000 plants

The triangle will have max area when it is isosceles. The altitude will be 4 [sqrt(5^-3^) =4]
So area = 4*6/2= 12 in^2