Related Rates Calculator Cone

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Related Rates Cone Calculator

Calculate dV/dt or dh/dt for a Conical Reservoir

Must be less than or equal to Total Height

What do you want to solve for?

Understanding Cone Related Rates

In calculus, related rates problems involve finding the rate at which one quantity changes by relating that quantity to others whose rates of change are known. For a cone, the volume $V$ is given by:

V = (1/3)πr²h

In a typical conical tank problem, the ratio of the radius $r$ to the height $h$ remains constant based on the dimensions of the container ($R/H$). Therefore, we can express the radius in terms of height:

r = (R/H) * h

The Related Rates Formula

By substituting $r$ into the volume formula, we get $V$ in terms of $h$ only. Differentiating with respect to time ($t$) using the chain rule gives us the relationship between $dV/dt$ and $dh/dt$:

dV/dt = π * (R/H)² * h² * (dh/dt)

Example Calculation

Suppose you have a cone with a total height of 10m and a base radius of 5m. If water is being pumped in at a rate of 2 m³/min ($dV/dt = 2$), how fast is the water level rising ($dh/dt$) when the water is 4m deep?

  1. Determine the ratio: $R/H = 5/10 = 0.5$.
  2. Use the formula: $2 = π * (0.5)² * (4)² * (dh/dt)$.
  3. Simplify: $2 = π * 0.25 * 16 * (dh/dt) \Rightarrow 2 = 4π * (dh/dt)$.
  4. Solve: $dh/dt = 2 / (4π) \approx 0.159$ m/min.
function toggleInputs() { var solveFor = document.querySelector('input[name="solveFor"]:checked').value; var dvGroup = document.getElementById('dv-input-group'); var dhGroup = document.getElementById('dh-input-group'); if (solveFor === 'dv') { dvGroup.style.display = 'block'; dhGroup.style.display = 'none'; } else { dvGroup.style.display = 'none'; dhGroup.style.display = 'block'; } } function calculateRelatedRates() { var H = parseFloat(document.getElementById('totalHeight').value); var R = parseFloat(document.getElementById('totalRadius').value); var h = parseFloat(document.getElementById('currentHeight').value); var solveFor = document.querySelector('input[name="solveFor"]:checked').value; var resultDiv = document.getElementById('calc-result'); var resultText = document.getElementById('result-text'); var resultSteps = document.getElementById('result-steps'); // Validation if (isNaN(H) || isNaN(R) || isNaN(h) || H <= 0 || R <= 0 || h H) { alert("Current liquid height cannot exceed total cone height."); return; } var k = R / H; // Constant ratio r/h var pi = Math.PI; var finalResult = 0; var formulaString = ""; if (solveFor === 'dv') { var dh_dt = parseFloat(document.getElementById('dh_dt').value); if (isNaN(dh_dt)) { alert("Please enter a valid dh/dt value."); return; } // dV/dt = PI * (R/H)^2 * h^2 * dh/dt finalResult = pi * Math.pow(k, 2) * Math.pow(h, 2) * dh_dt; resultText.innerHTML = "Rate of Volume Change (dV/dt): " + finalResult.toFixed(4) + " units³/time"; resultSteps.innerHTML = "Steps:1. Ratio (R/H) = " + k.toFixed(4) + "2. Formula: dV/dt = π * (" + k.toFixed(4) + ")² * (" + h + ")² * (" + dh_dt + ")" + "3. dV/dt = π * " + (k*k).toFixed(4) + " * " + (h*h) + " * " + dh_dt; } else { var dv_dt = parseFloat(document.getElementById('dv_dt').value); if (isNaN(dv_dt)) { alert("Please enter a valid dV/dt value."); return; } // dh/dt = (dV/dt) / (PI * (R/H)^2 * h^2) var denominator = pi * Math.pow(k, 2) * Math.pow(h, 2); finalResult = dv_dt / denominator; resultText.innerHTML = "Rate of Height Change (dh/dt): " + finalResult.toFixed(4) + " units/time"; resultSteps.innerHTML = "Steps:1. Ratio (R/H) = " + k.toFixed(4) + "2. Denominator (πr²) = π * (" + k.toFixed(4) + " * " + h + ")² = " + denominator.toFixed(4) + "3. dh/dt = " + dv_dt + " / " + denominator.toFixed(4); } resultDiv.style.display = 'block'; }

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