A quantity y varies inversely as the square of x. The difference between the value of y when x=6 and when x=10 is 16. Find the law connecting x and y
Let y and x be the variables representing the quantity and the independent variable, respectively. According to the given information, we can write the equation as:
y = k/x^2,
where k is a constant.
To find the value of k, we can use the given difference in values between y when x=6 and when x=10.
When x = 6, y = k/6^2 = k/36.
When x = 10, y = k/10^2 = k/100.
The difference between these two values is given as 16:
k/36 - k/100 = 16.
To solve this equation for k, we can find a common denominator:
(100k - 36k)/(36*100) = 16,
64k = 16 * 36 * 100,
k = (16 * 36 * 100)/64,
k = 9 * 9 * 100,
k = 8100.
Therefore, the law connecting x and y is:
y = 8100/x^2.
Step 1: Write down the given information.
We are told that the quantity y varies inversely as the square of x. This means that y is inversely proportional to x^2.
Step 2: Write down the formula for inverse variation.
The formula for inverse variation is y = k/x^2, where k is the constant of variation.
Step 3: Find the constant of variation.
To find the constant of variation, we can use the given information. We are told that the difference between the value of y when x=6 and when x=10 is 16.
Let's substitute these values into the formula:
y1 = k/6^2
y2 = k/10^2
The difference between y1 and y2 is 16, so we can write:
y2 - y1 = 16
Substituting the values:
(k/10^2) - (k/6^2) = 16
Step 4: Solve the equation.
To solve the equation, we need to find the value of k. Simplifying the equation:
k/100 - k/36 = 16
Common denominator is 900, so multiply both sides by 900:
9k - 25k = 14400
-16k = 14400
Divide both sides by -16:
k = -900
Step 5: Write down the final equation.
Now that we have found the value of k, we can substitute it into the formula:
y = -900/x^2
Therefore, the law connecting x and y is y = -900/x^2.