We will use geometric probability. The probability of a randomly chosen point in the grid being inside the circle is the ratio of the area of the circle to the area of the grid. The given coordinate grid is a square with a side length of a = 10. Let's find its area!
The area of the grid is 100. Let's now take a closer look at the circle.
As we can see, the circle has a radius of 2. Recall the formula for the area C of a circle with radius r.
C = π r^2
By substituting 2 for r into this formula, we can calculate the area of our circle.
The probability of the point being inside the circle is approximately 0.13.
b We are asked to find the probability of a point chosen at random in the given coordinate grid being inside the trapezoid.
Once again, we will use geometric probability. The probability of a randomly chosen point in the grid being inside the trapezoid is the ratio of the area of the trapezoid to the area of the grid. We already know from Part A that the area of the grid is 100. Let's now take a closer look at the trapezoid.
We can see that the trapezoid has a height h = 3 and bases of length b_1 = 2 and b_2 = 4. Remember the formula for the area T of a trapezoid with height h and bases of length b_1 and b_2.
T = 1/2 h ( b_1+ b_2)
By substituting 3 for h, 2 for b_1, and 4 for b_2 into this formula, we can calculate the area of our trapezoid.
The probability of the point being inside the trapezoid is 0.09.
c We are asked to find the probability of a point chosen at random in the given coordinate grid being inside the trapezoid, square, or circle. As before, we will use geometric probability.
The probability of a randomly chosen point in the grid being inside the trapezoid, square, or circle is the ratio of the sum of the areas of the trapezoid, square, and circle to the area of the grid. We know from Parts A and B that the area of the circle is approximately 12.57 and the area of the trapezoid is 9. Let's take a look at the square.
We can see that the side of the square is a hypotenuse of a right triangle with legs of length 2.
This tells us that the triangle is a 45^(∘)-45^(∘)-90^(∘) triangle. In this kind of triangle, the length of the hypotenuse is sqrt(2) times the length of the leg. In our case, the length of the hypotenuse is a = 2sqrt(2). Since the length of the hypotenuse of this triangle is the length of the side of our square, we can now calculate the area S of the square.
Now that we know that the area of the square is S = 8, we can calculate the combined area of all three figures. Remember that the area C of the circle is approximately 12.57 and the area T of the trapezoid is 9.
The combined area of the figures is approximately 29.57, and we know from Part A that the area of the grid is 100. We can now calculate the probability that a randomly chosen point in the grid is inside one of the figures.
P(point inside one of the figures) = Area of the Figures/Area of the Grid
