Find two numbers whose ratio is 4 to 3 and are such that the sum of their squares is 100.
let the numbers be 4x and 3x.
(4x)^2 + (3x)^2 = 100
16x^2 + 9x^2 = 100
25x^2 = 100
x^2 = 4
x = 2
So, the numbers are 6 and 8.
Check: 6^2 + 8^2 = 36+64 = 100
Or, you could have thought first of the ubiquitous 3-4-5 triangle.
8 to 6
To find two numbers with a ratio of 4 to 3 and whose sum of squares is 100, we can set up a system of equations. Let's assume the two numbers are represented by x and y.
According to the given ratio, we can write the equation x/y = 4/3.
Cross-multiplying, we get 3x = 4y.
Now, let's consider the sum of their squares. We have x^2 + y^2 = 100.
Since we know 3x = 4y, we can rewrite the equation as x = (4/3)y.
Substituting x in terms of y in the second equation, we have ((4/3)y)^2 + y^2 = 100.
Multiplying, we get (16/9)y^2 + y^2 = 100.
Combining like terms, we have (16/9 + 1) y^2 = 100.
Simplifying, we get (25/9) y^2 = 100.
Dividing both sides by (25/9), we have y^2 = (100 * 9) / 25.
y^2 = 36.
Taking the square root of both sides, we have y = ±6.
Now, we can substitute the value of y back into the x = (4/3)y equation to find x.
If y = 6, then x = (4/3) * 6 = 8.
If y = -6, then x = (4/3) * -6 = -8.
Therefore, the two numbers are 8 and 6, or -8 and -6, depending on the context of the problem.
To find two numbers whose ratio is 4 to 3, we can assume one number as 4x and the other as 3x. This means that the first number is four times larger than the second number.
Let's substitute these values into the second condition, where the sum of their squares is 100. We get:
(4x)^2 + (3x)^2 = 100
16x^2 + 9x^2 = 100
25x^2 = 100
To solve for x, we divide both sides of the equation by 25:
x^2 = 4
Taking the square root of both sides, we find:
x = ± 2
Now that we have the value of x, we can find the two numbers. Plugging x = 2 into our assumption, we have:
First number = 4x = 4(2) = 8
Second number = 3x = 3(2) = 6
Therefore, the two numbers are 8 and 6.