The following equation shows the approximate relationship between the percentage y of total personal consumption spent on food and the corresponding percentage x spent on education.
y = 38/x^0.38 percentage points; (6.5 ≤ x ≤ 17.5).
According to the model, spending on food is decreasing at a rate of ___?___ percentage points per one percentage point increase in spending on education when 10% of total consumption is spent on education.
Well, let me calculate that for you using my handy-dandy clown calculator.
Okay, so when x = 10, we can plug that into the equation and get:
y = 38/10^0.38
After some number crunching, the answer is approximately 0.400765 percentage points.
So, according to the model, spending on food is decreasing at a rate of approximately 0.400765 percentage points per one percentage point increase in spending on education when 10% of total consumption is spent on education.
Now that's some fancy math clowning around, isn't it?
To find the rate at which spending on food is decreasing per one percentage point increase in spending on education when 10% of total consumption is spent on education, we need to calculate the derivative of the equation with respect to x.
Taking the derivative of y = 38/x^0.38 gives us:
dy/dx = (-0.38 * 38) / (x^(0.38 + 1))
Simplifying further:
dy/dx = -14.44 / x^1.38
Now, we can substitute x = 10 to find the rate at which spending on food is decreasing when 10% of total consumption is spent on education.
dy/dx = -14.44 / 10^1.38
Calculating this value, we get:
dy/dx ≈ -0.428
Therefore, spending on food is decreasing at a rate of approximately 0.428 percentage points per one percentage point increase in spending on education when 10% of total consumption is spent on education.