Figure ABCD is translated down by 6 units:
1 answer:
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Answer:
The correct option is;
21 ft
Step-by-step explanation:
The equation of the parabolic arc is as follows;
y = a(x - h)² + k
Where the height is 25 ft and the span is 40 ft, the coordinates of the vertex (h, k) is then (20, 25)
We therefore have;
y = a(x - 20)² + 25
Whereby the parabola starts from the origin (0, 0), we have;
0 = a(0 - 20)² + 25
0 = 20²a + 25 → 0 = 400·a + 25
∴a = -25/400 = -1/16
The equation of the parabola is therefore;
To find the height 8 ft from the center, where the center is at x = 20 we have 8 ft from center = x = 20 - 8 = 12 or x = 20 + 8 = 28
Therefore, plugging the value of x = 12 or 28 in the equation for the parabola gives;
.
The approximate area of the region bounded by the curves f(x) = x / √(x² + 1) and g(x) = x⁴ - x is approximately 0.806.
<h3>How to determine the approximate area of the regions bounded by the curves</h3>
In this problem we must use definite integrals to determine the area of the region bounded by the curves. Based on all the information given by the graph attached below, the area can be defined in accordance with this formula:
A = A₁ + A₂ (1)
A₁ = ∫ [g(x) - f(x)] dx, for x ∈ [- 0.786, 0] (2)
A₂ = ∫ [f(x) - g(x)] dx, for x ∈ [0, 1.151] (3)
g(x) = x⁴ - x (4)
f(x) = x / √(x² + 1) (5)
Then, we proceed to find the integrals:
∫ g(x) dx = ∫ x⁴ dx - ∫ x dx = (1 / 5) · x⁵ - (1 / 2) · x² (6)
∫ f(x) dx = ∫ [x / √(x² + 1)] dx = (1 / 2) ∫ [2 · x / √(x² + 1)] dx = (1 / 2) ∫ [du / √u] = √u = √(x² + 1) (7)
And the complete expression for the integral is:
A = A₁ + A₂ (1b)
A₁ = (1 / 5) · x⁵ - (1 / 2) · x² - √(x² + 1), for x ∈ [- 0.786, 0] (2b)
A₂ = √(x² + 1) - (1 / 5) · x⁵ + (1 / 2) · x², for x ∈ [0, 1.151] (3b)
A₁ = 0.023
A₂ = 0.783
A = 0.023 + 0.783
A = 0.806
The approximate area of the region bounded by the curves f(x) = x / √(x² + 1) and g(x) = x⁴ - x is approximately 0.806.
To learn more on definite integral: brainly.com/question/14279102
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