A train leaves the station at $10:00$ am and travels due south at a speed of $60km/h$. Another train has been heading due west at $45km/h$ and reaches the same station at $11:00$ am. At what time were the two trains the closest together? (Preferably to be solved using derivatives!)
2026-04-03 06:49:36.1775198976
Calculus train question
4.3k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in CALCULUS
- Equality of Mixed Partial Derivatives - Simple proof is Confusing
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- Proving the differentiability of the following function of two variables
- If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Number of roots of the e
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- How to prove $\frac 10 \notin \mathbb R $
- Proving that: $||x|^{s/2}-|y|^{s/2}|\le 2|x-y|^{s/2}$
Related Questions in OPTIMIZATION
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- optimization with strict inequality of variables
- Gradient of Cost Function To Find Matrix Factorization
- Calculation of distance of a point from a curve
- Find all local maxima and minima of $x^2+y^2$ subject to the constraint $x^2+2y=6$. Does $x^2+y^2$ have a global max/min on the same constraint?
- What does it mean to dualize a constraint in the context of Lagrangian relaxation?
- Modified conjugate gradient method to minimise quadratic functional restricted to positive solutions
- Building the model for a Linear Programming Problem
- Maximize the function
- Transform LMI problem into different SDP form
Related Questions in PROBLEM-SOLVING
- How do you prevent being lead astray when you're working on a problem that takes months/years?
- How to prove the inequality $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}\geq \log (2)$?
- How to solve higher order polynomial equations?
- Methods in finding invariant subspaces?
- Question about the roots of a complex polynomial
- Using a counting argument to prove some equalities? (Problem Solving)
- (Problem Solving) Proving $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\frac{1}{k+m+1}=\sum_{k=0}^{m}(-1)^k\binom{m}{k}\frac{1}{k+n+1}$
- (Problem Solving) Proving $|x|^p +|y|^p \geq |x+y|^p$
- Each vertex of the square has a value which is randomly chosen from a set.
- Fill in the blanks
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Observe that you want the length of the hypotenuse of a right triangle with vertical leg (the southern bound train) of length of $\;60x\;$, and a horizontal leg (the other train) of length $\;45(1-x)\;$ , with $\;x\,$= time ellapsed since $\;10:00\;$ , so the wanted hypotenuse's length (squared) is
$$f(x)=3,600 x^2+2,025(1-x)^2=5,625x^2-4,050 x+2,025=0\implies$$
$$f'(x)=11,250x-4,050=0\implies x=\frac{4,050}{11,250}=\frac25\implies$$
At $\;10:00 + 24'=10:24\;$ the distance was minimal, and this was
$$\sqrt{f\left(\frac25\right)}\approx36.12 \;kms.$$