The dimensions of the rectangle are 26 × 8.
In the question, we are asked to find the dimensions of the rectangle whose base exceeds three times its height by 8, and its semi-perimeter is 32.
For a rectangle, we know that the perimeter is given as,
Perimeter = 2(Base + Height).
Now, the semi-perimeter is half of the perimeter, that is, (1/2)*Perimeter.
Thus, semi-perimeter = (1/2)*Perimeter = (1/2)*2(Base + Height) = Base + Height.
We assume the height of the rectangle to be x.
Thus, the base can be shown as 8 more than three times the height, that is, the base = 3x + 8.
Now, we have base = 3x + 8, height = x, and the semi-perimeter = 32.
Substituting these values in the formula for the semi-perimeter,
Semi-perimeter = Base + Height, we get:
32 = (3x + 8) + x,
or, 32 = 4x + 8,
or, 4x = 32 - 8,
or, 4x = 24,
or, x = 24/4,
or, x = 6.
Thus, we have height = x = 6, and
the base = 3x + 8 = 3(6) + 8 = 18 + 8 = 26.
Thus, the dimensions of the rectangle are 26 × 8.
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