Distance is a measurement. Distance refers to the length between two points or between two objects. For example, you might need the distance between your hometown and another city so you can calculate how much gas you may need for a trip. Likewise, you may need the distance between two walls to determine how much paint you need to paint a room. Sometimes distance, however, is not referred to as distance in a problem, but perhaps the length, width, height, or how far are used in place of the word distance.
You can calculate the distance in a couple of ways. One way is to use the rate of what you are calculating and multiply it by the time it takes. This distance formula is written as {eq}d=rt {/eq}. The other way to calculate distance is to use the coordinate plane. This distance equation is written as {eq}d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} {/eq}. In this equation, you will need to have the (x, y) coordinate pair for two points to calculate the distance.
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Try it nowA coordinate provides a location of a point in the coordinate plane. Each coordinate has two numbers in it separated by a comma and placed in parentheses like the following example: {eq}(1, 2) {/eq}. The first number represents the x value of the coordinate, and the second number represents the y value of the coordinate.
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To calculate the distance between the two points shown in the example above, you will need the distance equation (also referred to as distance formula in geometry) {eq}d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} {/eq} and you will need the coordinates of two points. The two points in the example above are {eq}(1,2) {/eq} and {eq}(-1,-6) {/eq}. In the formula, you will notice that there are subscripts 1 and 2. These subscripts are just telling you that you have a first and second x value and a first and second y value. Another way to think about it is that you have two points. You have a first point and a second point. It does not matter which point you call the first and which you call the second. So, {eq}(1,2) {/eq} can be considered the first or second point when using the distance equation. And, {eq}(-1,-6) {/eq} can be considered the first or second point when using the distance equation.
Using the example mentioned above, here is how the distance equation would work if you use {eq}(1,2) {/eq} as the first point.
{eq}d=\sqrt{(1+1)^2+(2+6)^2} {/eq}
{eq}d=\sqrt{2^2+8^2} {/eq}
{eq}d=\sqrt{4+64}=\sqrt{68}\approx8.24621 {/eq}
Here is how the distance equation would work if you use {eq}(-1,-6) {/eq} as the first point.
{eq}d=\sqrt{(-1-1)^2+(-6-2)^2} {/eq}
{eq}d=\sqrt{(-2)^2+(-8)^2} {/eq}
{eq}d=\sqrt{4+64}=\sqrt{68}\approx8.24621 {/eq}
You can see that it does not matter which point you identify as the first and identify as the second, as you will get the same answer for both.
So far, the distance equation has been discussed in regards to a two-dimensional plane. If however, you would like to use the distance equation in three-dimensional problems, you will have to use the following equation: {eq}d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} {/eq}.
Given the following image, calculate the distance between the two points.
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The two points in the image are {eq}(4,3) {/eq} and {eq}(-2, 8) {/eq}.
Plug these two coordinates into the distance formula.
{eq}d=\sqrt{(-2-4)^2+(8-3)^2} {/eq}
{eq}d=\sqrt{(-6)^2+(5)^2} {/eq}
{eq}d=\sqrt{36+25} {/eq}
{eq}d=\sqrt{61}\approx7.8102 {/eq}
The distance between the two coordinates is approximately {eq}7.8102 {/eq} units.
In this example, we will look at how to use the distance formula in a three-dimensional situation.
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The coordinates in this example are {eq}(1,2,3) {/eq} and {eq}(-4,5,-7) {/eq}.
Plug these two coordinates into the distance formula.
{eq}d=\sqrt{(-4-1)^2+(5-2)^2+(-7-3)^2} {/eq}
{eq}d=\sqrt{(-5)^2+(3)^2+(-10)^2} {/eq}
{eq}d=\sqrt{25+9+100} {/eq}
{eq}d=\sqrt{134}\approx11.5758 {/eq}
The distance between these two coordinates is approximately {eq}11.5758 {/eq} units.
When given a real-world situation that involves rates and time, there is another way to calculate the distance. Use the distance rate time formula :{eq}d=rt {/eq}.
In this formula, you must make sure that you are using the same units of measure for the rate and the time. For instance, if you have a rate of {eq}50 {/eq} miles per day, then your time must also be in days.
Denise has been riding her bike for a few weeks now. She is building her stamina to participate in a local event in which the route is a total distance of {eq}16 {/eq} miles. She knows that she can complete {eq}5 {/eq} miles in one hour. The event is scheduled to take place over {eq}4 {/eq} hours. Denise wants to know how far she would travel in {eq}4 {/eq} hours if she bikes at her current rate to make sure she can finish the event in the allotted time. Calculate her distance over the {eq}4 {/eq} hours. Use the following formula.
{eq}d=rt {/eq}
Replace {eq}r {/eq} - the rate - with Denise's rate of {eq}5 {/eq} miles per hour and {eq}t {/eq} - the time - with {eq}4 {/eq}.
{eq}d=(5)(4) {/eq}
{eq}d=20 {/eq}
Denise will travel {eq}20 {/eq} miles in the {eq}4 {/eq} hours of the event at her current pace. She will be able to complete the event in the time allotted as she can travel {eq}20 {/eq} miles in {eq}4 {/eq} hours and the route is only {eq}16 {/eq} miles long.
Miseal is in the market for a new vehicle. He wants to purchase one with great gas mileage. In his research, he finds that the Toyota Prius' gas mileage is {eq}54 {/eq} miles per gallon on the highway. He wonders how far he could travel on {eq}16 {/eq} gallons of gas. He uses the following formula {eq}d=rt {/eq} to find the total distance he can travel on {eq}16 {/eq} gallons of gas if he can travel {eq}54 {/eq} miles per gallon.
{eq}d=rt {/eq}
Substitute {eq}54 {/eq} for {eq}r {/eq} and {eq}16 {/eq} for {eq}t {/eq}.
{eq}d=(54)(16) {/eq}
{eq}d=864 {/eq}
Miseal found he could travel {eq}864 {/eq}miles on {eq}16 {/eq} gallons of gas.
You can also use the distance formula, {eq}d=rt, {/eq} to find rate and time.
You can use the distance formula, {eq}d=rt {/eq}, to determine the miles per hour you are traveling. To determine miles per hour, you will need to rearrange the formula and have the formula equal {eq}r {/eq}.
{eq}d=rt {/eq}
Divide both sides by {eq}t {/eq} so {eq}r {/eq} will be isolated
{eq}\frac{d}{t}=r {/eq}
You can now use this rewritten formula to calculate miles per hour.
If you travel a distance of {eq}345 {/eq} miles in {eq}5 {/eq} hours and want to know how many miles per hour you traveled, replace {eq}d {/eq} with {eq}345 {/eq} and {eq}t {/eq} with {eq}5 {/eq}.
{eq}\frac{345}{5}=r {/eq}
{eq}r=69 {/eq}
Your rate of travel will be {eq}69 {/eq} miles per hour if you travel {eq}345 {/eq} miles in {eq}5 {/eq} hours.
You can also use the distance formula, {eq}d=rt {/eq}, to determine the time it will take you to travel. To determine the time, you will need to rearrange the formula and have the formula equal {eq}t {/eq}.
{eq}d=rt {/eq}
Divide both sides by {eq}r {/eq} so {eq}t {/eq} will be isolated.
{eq}\frac{d}{r}= t {/eq}
You can now use this rewritten formula to calculate the time it will take to travel.
If you have a total distance of {eq}625 {/eq} miles to travel and travel at a rate of {eq}65 {/eq} miles per hour, replace {eq}d {/eq} with {eq}625 {/eq} and {eq}r {/eq} with {eq}65 {/eq} to determine the time it will take you to travel the total {eq}625 {/eq} miles.
{eq}\frac{625}{65}=t {/eq}
{eq}9.61 \approx t {/eq}
It will take you approximately {eq}9.6 {/eq} hours to travel {eq}625 {/eq} miles when you are traveling at a rate of {eq}65 {/eq} miles per hour.
Distance can be calculated in various ways. One way is to use the distance equation {eq}d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} {/eq}. This equation is used if you are given coordinates in the coordinate plane. It can be used for two-dimensional and three-dimensional coordinates. Another way to calculate distance is to use the distance rate time formula. This formula is {eq}d=rt {/eq}. The distance rate time formula is generally used in distance word problems, and you must remember to keep the units of the rate and the time the same when calculating the distance. The distance rate time formula can also be rearranged to solve for a rate or time if you need to find one or the other.
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You calculate distance traveled by using the formula d=rt. You will need to know the rate at which you are traveling and the total time you traveled. You can then multiply these two numbers together to determine the distance traveled.
The distance formula is found by solving the Pythagorean Theorem for c. The Pytahgorean Theorem is a^2+b^2=c^2. Once you have solved this for c, you will have the distance formula.
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