Gateway Arch in St. Louis was constructed using the equation
y=211.49-20.96cosh(0.03291765x)
where |x|<=91.20
a) what is the height at its center in meters
b) At what points is height 70m (2 values of x)
c) what is the slope of arch at points in part b
a) well, cosh(0) = 1
b)
211.49-20.96cosh(0.03291765x) = 70
20.96cosh(0.03291765x) = 141.49
cosh(0.03291765x) = 6.75
x = arccosh(6.75)/0.03291765
c)
y' = -(20.96)(0.03291765x)sinh(0.03291765x)
so, just plug in the values from (b).
Use the graphs's symmetry to eliminate one calculation.
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To answer these questions, let's go step-by-step:
a) Finding the height at the center:
To find the height at the center, we need to find the value of y when x is 0. Plug in x = 0 into the equation:
y = 211.49 - 20.96cosh(0.03291765 * 0)
y = 211.49 - 20.96cosh(0)
Since cosh(0) equals 1, the equation simplifies to:
y = 211.49 - 20.96 * 1
y = 211.49 - 20.96
y = 190.53 meters
Therefore, the height at the center of the Gateway Arch in St. Louis is 190.53 meters.
b) Finding the points where the height is 70 meters:
To find the points where the height is 70 meters, we need to solve the equation for y = 70. Plug in y = 70 into the equation:
70 = 211.49 - 20.96cosh(0.03291765x)
To solve this equation, we need to isolate the cosh function. Rearrange the equation as follows:
20.96cosh(0.03291765x) = 211.49 - 70
20.96cosh(0.03291765x) = 141.49
Divide both sides of the equation by 20.96:
cosh(0.03291765x) = 6.74
To solve for x, we need to use the inverse hyperbolic cosine function (acosh). Take the acosh of both sides:
0.03291765x = acosh(6.74)
Now divide both sides by 0.03291765 to solve for x:
x = acosh(6.74) / 0.03291765
Using a calculator, we find that x is approximately 80.31 or -80.31 meters.
Therefore, the two values of x where the height is 70 meters are approximately 80.31 and -80.31 meters.
c) Finding the slope of the arch at the points in part b:
To find the slope of the arch at the points where the height is 70 meters, we need to find the derivative of the equation with respect to x. The derivative represents the slope.
Differentiate the equation y = 211.49 - 20.96cosh(0.03291765x) with respect to x:
dy/dx = -20.96 * sinh(0.03291765x) * 0.03291765
Evaluate the derivative at the points x = 80.31 and x = -80.31:
At x = 80.31:
dy/dx = -20.96 * sinh(0.03291765 * 80.31) * 0.03291765
At x = -80.31:
dy/dx = -20.96 * sinh(0.03291765 * -80.31) * 0.03291765
Calculating these values will give you the slopes at the respective points.
Note: The values obtained for the slopes will be in meters per unit of x.
To answer these questions, we will need to understand the provided equation and how to analyze it.
The given equation for the Gateway Arch is y = 211.49 - 20.96cosh(0.03291765x), where |x| <= 91.20.
a) The height at the center of the arch can be found by substituting x = 0 into the equation. So, we have:
y = 211.49 - 20.96cosh(0.03291765 * 0)
y = 211.49 - 20.96cosh(0)
y = 211.49 - 20.96 * 1
y = 211.49 - 20.96
y ≈ 190.53 meters
Therefore, the height at the center of the arch is approximately 190.53 meters.
b) To find the points at which the height is 70 meters, we need to solve the equation y = 70:
70 = 211.49 - 20.96cosh(0.03291765x)
To solve for x, we need to rearrange the equation:
20.96cosh(0.03291765x) = 211.49 - 70
20.96cosh(0.03291765x) = 141.49
Now we can isolate the cosh term by dividing both sides by 20.96:
cosh(0.03291765x) = 141.49 / 20.96
Next, inverse cosh both sides to solve for x:
0.03291765x = acosh(141.49 / 20.96)
Divide by 0.03291765 to solve for x:
x = acosh(141.49 / 20.96) / 0.03291765
Now, use a calculator to evaluate the expression on the right side to find the two values of x.
c) To find the slope of the arch at the points found in part b, we need to differentiate the equation y = 211.49 - 20.96cosh(0.03291765x) with respect to x.
The derivative of cosh(u) is sinh(u), and the derivative of constant * u is the constant. So, we have:
y' = -20.96 * sinh(0.03291765x) * 0.03291765
Evaluate this expression for each of the x-values found in part b to obtain the slopes at those points.