The likelihood that a child will attend a live musical performance can be modeled by the following equation.
q=0.01(0.0007 x**2+0.4 x +33) text( ) \(15<=x<=100\)
Here, q is the fraction of children with annual household income x thousand dollars who will attend a live musical performance during the year. Compute the income elasticity E of demand at an income level of $30,000. (Round your answer to two decimal places.)
To compute the income elasticity of demand (E), we need to use the formula:
E = (dq/dx) * (x/q)
where dq/dx is the derivative of q with respect to x, evaluated at a specific income level, and x is the income level.
First, let's find the derivative of q with respect to x:
dq/dx = 0.01(0.0007(2x) + 0.4)
Next, we can substitute the income level x = $30,000 into the derivative:
dq/dx = 0.01(0.0007(2 * 30000) + 0.4)
dq/dx = 0.01(0.0007(60000) + 0.4)
dq/dx = 0.01(42 + 0.4)
dq/dx = 0.01(42.4)
dq/dx = 0.424
Now, let's substitute the values of dq/dx and x into the income elasticity formula:
E = (dq/dx) * (x/q)
E = (0.424) * (30000 / q)
To find the value of q at an income level of $30,000, we can substitute it into the equation for q:
q = 0.01(0.0007 x^2 + 0.4 x + 33)
q = 0.01(0.0007(30000^2) + 0.4(30000) + 33)
q ≈ 0.8617
Finally, we can substitute the values of q and x into the income elasticity formula:
E = (0.424) * (30000 / 0.8617)
E ≈ 487.90
Therefore, the income elasticity of demand at an income level of $30,000 is approximately 487.90.
To compute the income elasticity E of demand at an income level of $30,000, we need to find the derivative of q with respect to x and then evaluate it at x = 30.
First, let's find the derivative of q with respect to x. To do this, we differentiate each term of the equation separately.
Given: q = 0.01(0.0007x^2 + 0.4x + 33)
Let's differentiate the first term: 0.0007x^2
The derivative of 0.0007x^2 with respect to x is 0.0014x.
Next, let's differentiate the second term: 0.4x
The derivative of 0.4x with respect to x is 0.4.
Finally, let's differentiate the constant term: 33
Since it's a constant, its derivative is zero.
Now, let's substitute the derivatives back into the equation:
dq/dx = 0.01(0.0014x + 0.4)
Next, let's evaluate the derivative at x = 30:
dq/dx = 0.01(0.0014(30) + 0.4)
= 0.01(0.042 + 0.4)
= 0.01(0.442)
= 0.00442
Now, let's compute the income elasticity E using the formula:
E = (dq/dx) * (x/q)
Substituting the values:
E = (0.00442) * (30,000 / q)
Since the value of q is not given, we cannot calculate the exact income elasticity E. We require the value of q for the given income level to compute it.