A plot of ground in the shape of a circular sector (a wedge of pie) is to have a border of roses along the straight lines and tulips along the circular arc. Roses cost 20$/meter; tulips cost 15$/meter. If the area of the plot is to be 100 square meters, what is the least the flower can cost?

Let the radius of the circular sector be r meters and the central angle be \theta radians.

The area of a circular sector is given by the formula:
A = \frac{1}{2}r^2\theta

Given that the area of the plot is 100 square meters, we have:
100 = \frac{1}{2}r^2\theta

Now, the perimeter of the circular sector consists of two parts - the straight lines (of length 2r each) and the circular arc (of length r\theta ). The cost of roses along the straight lines is 20 /meter and the cost of tulips along the circular arc is 15 /meter. Therefore, the total cost C is given by:
C = 40r + 15r\theta

Substitute \theta = \frac{200}{r^2} from the area equation into the cost equation:
C = 40r + 15r\left(\frac{200}{r^2}\right) = 40r + \frac{3000}{r}

To minimize the cost, we differentiate C with respect to r and set the derivative equal to zero:
\frac{dC}{dr} = 40 - \frac{3000}{r^2}
\frac{dC}{dr} = 0 \Rightarrow 40 = \frac{3000}{r^2}
r^2 = \frac{3000}{40} = 75
r = \sqrt{75} = 5\sqrt{3} \text{ meters}

Substitute r = 5\sqrt{3} back into the cost equation to find the minimum cost:
C = 40(5\sqrt{3}) + \frac{3000}{5\sqrt{3}}
C=200\sqrt{3}+200\sqrt{3}=400\sqrt{3}\approx\$692.82

Therefore, the least the flowers can cost is $692.82.