Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. We are going to give several forms of the heat equation for reference purposes, but we will only be really solving one of them.

A snowball is melting at a rate of 1 cubic inch per minute. At what rate is the radius changing when the snowball has a radius of 2 inches? Note that since volume and radius are changing, we have to use variables for them. Define what we have and what we want when. Determine what equation we need: Make sure that all variables will differentiate to a rate we either have or need: Differentiate with respect to time: Plug in what we know at the time we know it, and solve for what we need: It makes sense that it is negative, since radius is decreasing along with the volume.

A foot ladder is resting against a wall, and its base is slipping on the floor away from the wall at a rate of 2 feet per minute. Find the rate of change of the height of the ladder at the time when the base is 20 feet from the base of the wall.

The ladder slanted will be the hypotenuse of a right triangle formed by the wall, the floor, and the ladder. Now plug everything in: It makes sense that it is negative, since the ladder is slipping down the wall. Here are more problems: Related Rates Problem Steps and Solution Water is being poured into a cylindrical can that is 20 inches in height and has a radius of 8 inches.

The water is being poured at a rate of 3 cubic inches per second. How fast is the height of the water in the can changing when the height is 8 feet deep?

Note that since the radius and height of the actual can is not changing, we can use constants for them. Since the height and volume of the water inside is changing, we have to use variables: We need to relate the radius and height of a cylinder to its volume: We now have everything we need!

It makes sense that it is positive, since the can is filling up. Related Rates Problem Steps and Solution A woman 5 feet tall walks at a rate of 6 feet per second away from a lamppost. When the woman is 6 feet from the lamppost, her shadow is 8 feet tall.

At what rate is the length of her shadow changing when she is 12 feet from the lamppost? At what rate is the tip of her shadow changing when she is 12 feet from the lamppost?

We need variables for both of these distances, since they are changing. Now this is tricky: This is because we need to measure the tip of the shadow in reference to the base of the lamppost. This is the rate of the length of the shadow. Related Rates Problem Steps and Solution A hot air balloon is rising straight up and is tracked by a range finder feet from point of lift-off.

How fast is the balloon rising at that time? Note that since height and the angle of elevation are changing, we have to use variables for them. We also know the distance on the ground from the range finder to the balloon is We need to relate the height and base of the right triangle to the angle of elevation; we can use trigonometry: Yes, radians per minute is the rate the angle of elevation is changing.

It makes sense that it is positive since the angle is increasing along with the height. Understand these problems, and practice, practice, practice! Click on Submit the arrow to the right of the problem to solve this problem.

You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems. You can even get math worksheets. There is even a Mathway App for your mobile device.

Welcome to She Loves Math! And, even better, a site that covers math topics from before kindergarten through high school.Referencing subordinate equations can be done using either of two methods: adding a label after the \begin {subequations} command, which will reference the main equation ( above), or adding a label at the end of each line, before the \\ command, which will reference the sub-equation (a or b above).

It is possible to add both labels in case both types of references are needed. IXL Math On IXL, math is more than just numbers.

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With unlimited questions, engaging item types, and real-world scenarios, IXL helps learners experience math at its most mesmerizing! The first Figure is the FEA mesh detail in the unloaded condition. The second Figure is the deformations that occur at , rpm just before failure with the deformations amplified by 5X.

Introduction. The shortest path between two given points in a curved space, assumed to be a differential manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of ashio-midori.com has some minor technical problems, because there is an infinite.

Introduction. The shortest path between two given points in a curved space, assumed to be a differential manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of ashio-midori.com has some minor technical problems, because there is an infinite.

The first Figure is the FEA mesh detail in the unloaded condition. The second Figure is the deformations that occur at , rpm just before failure with the deformations amplified by 5X.

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