Enter the known components of an inverse relationship to find the unknown.
y = k / x (Inverse Proportionality)
y = m*x + c (Linear with negative slope)
y = a * exp(-b*x) (Exponential Decay)
The calculated unknown value is:
Understanding Inverse Equations and Their Applications
An inverse equation describes a relationship between two variables where as one variable increases, the other variable decreases proportionally, or where the relationship can be modeled with a negative slope or decay. This is in contrast to direct relationships where both variables change in the same direction. Inverse relationships are fundamental in various scientific, economic, and engineering fields.
Types of Inverse Relationships
This calculator focuses on three common types of inverse relationships:
1. Inverse Proportionality (y = k / x)
This is the classic form of inverse variation. The product of the two variables is a constant (k). As 'x' gets larger, 'y' gets smaller, and vice-versa. The graph of this relationship is a hyperbola.
Use Cases:
Physics: For a fixed amount of gas at constant temperature, pressure (P) is inversely proportional to volume (V) (Boyle's Law: P = k/V).
Economics: The price of a good and the quantity demanded can sometimes exhibit inverse proportionality, though demand curves are more complex.
Engineering: Resistance (R) in a simple circuit might be related to other factors inversely.
Calculation Example: If pressure (y) is inversely proportional to volume (x) with a constant k = 100, and the current volume is 20 units (x), what is the pressure (y)?
y = k / x = 100 / 20 = 5. So, the pressure is 5 units.
If we know y=5 and k=100, we can find x: x = k / y = 100 / 5 = 20.
2. Linear Relationship with Negative Slope (y = mx + c)
While not strictly "inverse" in the same way as proportionality, a linear equation with a negative slope ('m' < 0) represents a decreasing relationship. As 'x' increases, 'y' decreases.
Use Cases:
Economics: The relationship between the price of a product and the quantity a supplier is willing to produce (supply curve, often simplified).
Finance: The relationship between risk and return in some simplified models.
Modeling Trends: Many real-world phenomena show a general decreasing trend over time or with increasing input.
Calculation Example: A company's profit (y) decreases linearly with the number of marketing complaints (x) according to y = -2x + 50. If there are 10 complaints (x), what is the profit (y)?
y = -2 * 10 + 50 = -20 + 50 = 30. The profit is 30 units.
If the profit is 40 units (y), how many complaints (x) were there? 40 = -2x + 50 => -10 = -2x => x = 5 complaints.
3. Exponential Decay (y = a * exp(-b*x))
This describes a process where a quantity decreases at a rate proportional to its current value. The rate of decrease slows down over time. 'a' is the initial value, 'b' is the decay rate constant, and 'exp' is the natural exponential function (e).
Use Cases:
Physics: Radioactive decay, cooling of an object.
Biology: Drug concentration in the bloodstream over time.
Finance: Depreciation of assets.
Calculation Example: The concentration of a drug (y) in the body decreases exponentially with time (x) according to y = 100 * exp(-0.2*x). What is the concentration after 5 hours (x)?
y = 100 * exp(-0.2 * 5) = 100 * exp(-1) ≈ 100 * 0.3678 ≈ 36.78 units.
If the initial concentration is 100 (a) and after some time (x) the concentration is 10 (y), what is the decay rate (b)? (This requires logarithms to solve for b, which this calculator simplifies by letting you input b).
This calculator helps solve for one unknown variable when the others and the type of relationship are known, providing a quick tool for understanding these fundamental mathematical concepts.
function updateEquation() {
var type = document.getElementById('equationType').value;
document.getElementById('inverseProportionalityInputs').style.display = (type === 'inverse_proportionality') ? 'block' : 'none';
document.getElementById('linearInverseInputs').style.display = (type === 'linear_inverse') ? 'block' : 'none';
document.getElementById('exponentialDecayInputs').style.display = (type === 'exponential_decay') ? 'block' : 'none';
// Clear previous results and input values when changing type
document.getElementById('result').style.display = 'none';
document.getElementById('k_inv').value = ";
document.getElementById('x_inv').value = ";
document.getElementById('y_inv').value = ";
document.getElementById('m_lin').value = ";
document.getElementById('x_lin').value = ";
document.getElementById('y_lin').value = ";
document.getElementById('c_lin').value = ";
document.getElementById('a_exp').value = ";
document.getElementById('b_exp').value = ";
document.getElementById('x_exp').value = ";
document.getElementById('y_exp').value = ";
}
function calculate() {
var type = document.getElementById('equationType').value;
var resultValueElement = document.getElementById('resultValue');
var resultDiv = document.getElementById('result');
var calculatedValue = ";
var validInputs = true;
if (type === 'inverse_proportionality') {
var k = parseFloat(document.getElementById('k_inv').value);
var x = parseFloat(document.getElementById('x_inv').value);
var y = parseFloat(document.getElementById('y_inv').value);
if (!isNaN(k) && !isNaN(x) && x !== 0 && isNaN(y)) { // Solve for y
calculatedValue = k / x;
} else if (!isNaN(k) && !isNaN(y) && y !== 0 && isNaN(x)) { // Solve for x
calculatedValue = k / y;
} else if (!isNaN(x) && !isNaN(y) && y !== 0 && isNaN(k)) { // Solve for k
calculatedValue = x * y;
} else {
validInputs = false;
alert("Please provide two known values and leave one blank to solve for it. Ensure k and y are not zero when solving for x.");
}
} else if (type === 'linear_inverse') {
var m = parseFloat(document.getElementById('m_lin').value);
var x = parseFloat(document.getElementById('x_lin').value);
var y = parseFloat(document.getElementById('y_lin').value);
var c = parseFloat(document.getElementById('c_lin').value);
if (!isNaN(m) && !isNaN(x) && isNaN(y) && !isNaN(c)) { // Solve for y
calculatedValue = (m * x) + c;
} else if (!isNaN(m) && !isNaN(y) && isNaN(x) && !isNaN(c)) { // Solve for x
if (m === 0) {
validInputs = false;
alert("Cannot solve for x when slope (m) is zero and y is known.");
} else {
calculatedValue = (y – c) / m;
}
} else if (!isNaN(m) && !isNaN(x) && isNaN(c) && !isNaN(y)) { // Solve for c
calculatedValue = y – (m * x);
} else if (!isNaN(x) && !isNaN(y) && isNaN(m) && !isNaN(c)) { // Solve for m
if (x === 0) {
validInputs = false;
alert("Cannot solve for m if x is zero and c is known.");
} else {
calculatedValue = (y – c) / x;
}
}
else {
validInputs = false;
alert("Please provide three known values and leave one blank to solve for it.");
}
} else if (type === 'exponential_decay') {
var a = parseFloat(document.getElementById('a_exp').value);
var b = parseFloat(document.getElementById('b_exp').value);
var x = parseFloat(document.getElementById('x_exp').value);
var y = parseFloat(document.getElementById('y_exp').value);
if (!isNaN(a) && !isNaN(b) && !isNaN(x) && isNaN(y)) { // Solve for y
calculatedValue = a * Math.exp(-b * x);
} else if (!isNaN(a) && !isNaN(b) && !isNaN(y) && isNaN(x)) { // Solve for x
if (a === 0 || y / a <= 0) {
validInputs = false;
alert("Cannot solve for x: Ensure a is not zero and y/a is positive.");
} else {
calculatedValue = -Math.log(y / a) / b;
}
} else if (!isNaN(a) && !isNaN(x) && !isNaN(y) && isNaN(b)) { // Solve for b
if (a === 0 || y / a <= 0) {
validInputs = false;
alert("Cannot solve for b: Ensure a is not zero and y/a is positive.");
} else {
calculatedValue = -Math.log(y / a) / x;
}
} else if (!isNaN(b) && !isNaN(x) && !isNaN(y) && isNaN(a)) { // Solve for a
if (Math.exp(-b * x) === 0) {
validInputs = false;
alert("Cannot solve for a: The exponential term is zero.");
} else {
calculatedValue = y / Math.exp(-b * x);
}
}
else {
validInputs = false;
alert("Please provide three known values and leave one blank to solve for it. Ensure relevant values (like 'a' or 'y/a') are not zero or negative where applicable.");
}
}
if (validInputs && calculatedValue !== '') {
resultValueElement.textContent = calculatedValue.toFixed(4); // Display with 4 decimal places
resultDiv.style.display = 'block';
} else if (validInputs && calculatedValue === '') {
// This case is for when no calculation was performed due to insufficient inputs, already handled by alert.
}
}