Let L be the length of the rectangular body, W be the width of the rectangular body,

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Question:

Let L be the length of the rectangular body, W be the width of the rectangular body, and r be the radius of each semicircle. The objective is to maximize the volume of the toy car, which is given by:

V = (1/2)?r^2L + ?r^2W^2

Subject to the following constraints:

2L + 2r + ?r = 50 (total wire constraint)
r ? 0, L ? 0, W ? 0 (non-negativity constraints)

We can solve the total wire constraint for L, which gives:

L = (50 - 2r - ?r) / 2

Substituting this into the volume constraint, we get:

V = (1/2)?r^2[(50 - 2r - ?r) / 2] + ?r^2W^2

Expanding and simplifying, we get:

V = (1/4)?r^2(50 - 2r - ?r) + ?r^2W^2
V = (1/4)?r^2(50 - ?r) + ?r^2W^2

To find the maximum volume, we take the derivative of V with respect to r and set it equal to zero:

dV/dr = (1/4)?(2r - ?)(50 - 2?r) + 2?rW^2 = 0

Solving for r, we get:

r = (25 - ?W^2) / (2 - ?)

Substituting this value of r into the total wire constraint, we can solve for W:

2L + 2r + ?r = 50
L = (25 - r - (?/2)r) / 2
W = sqrt[(50 - 2r - 2L) / ?]

Substituting the value of r into these equations, we get:

r ? 3.57 cm, L ? 12.27 cm, W ? 7.98 cm

Therefore, the dimensions of the rectangular body should be approximately 12.27 cm by 7.98 cm, and the radii of the semicircles should be approximately 3.57 cm. With these dimensions, the maximum volume of the toy car is approximately:

V ? 559.02 cubic cm.