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Grade 7/8 POTW Firstly, I will list some things we know. - Time 1:15 - Same starting point - Gail Storm North 24 km/h - Dusty Storm East 32 km/h - When will there be a distance of 130 km. Firstly, if we draw lines to show how far they are from each other as well as the starting point, you form a right triangle with the hypotenuse being the distance away from each other. This is probably the right track (ha ha) since this is the geometry unit as well. Anyways a^2 + b^2 = c^2. a = Gail Storm traveling distance, b = Dusty Storm travelling distance, c = distance from each other. From here we just see how large c^2. Right now, c = 130 km. 130^2 = 16900. Now, instead of trial and error, I'll find the ratio of Dusty and Gail's speeds as this can find the answer immediately. Gail to Dusty = 24:32 = 3:4. However, this is without squaring it which means we have to square each number. 24^2 = 576, 32^2 = 1024. 576:1024 = 9:16. 9:16 of 16900. This is one way to continue from here. 9 + 16 = 25. 16900/25 = 676. 676 x 9 = 6084. 676 x 16 = 10816. The square root of 6084 is 78. The square root of 10816 is 104. Now we divide. 78/24 = 3.25. 104/32 = 3.25. They are both the same which proves this is most likely correct. 104 x 0.75 (3/4) = 78. More proof it is correct as it suits the ratio from before. 3.25 hours = 3 hours and 15 minutes (.25 = 1/4 = 1/4 of 60 = 15). 1:15 + 3 hours and 15 minutes = 4:30. They will have a distance of 130 km at 4:30. Dusty Storm traveled 104 km while Gail Storm traveled 78 km.
POTW: Info: - Gail- 24 km/h- north - Dusty- 32 km/h- east - What time will they be 130 km apart? And how far will they have travelled?
Since both of them are travelling north and east (assuming that they do not change directions or slant slightly), they are riding two of the lengths of the triangle. The slant would be the distance between the two bikers. Using the Pythagorean Theorom, we can reverse it to find the other two values. a^2 + b^2 = c^2. a^2 + b^2 = 130^2 a^2 + b^2 = 16900 Using x as the multiple: (24t)^2 + (32t)^2 = 130^2 (576t)^2 + (1024)^2 = 16900 (1600t) ^2 = 16900 (16t)^2= 169 t^2= 169/16 (this cannot be divided) Simplify 13/4=3.25 Since we are dealing with 60 minutes is a whole: 3 hours 15 min t= 13/4 24 x 13/4 = 78 = Gail 32 x 13/4 = 104 = Dusty 1:!5 + 3 h 15 min = 4:30 pm. (The dedication to bike for 3 hours plus...)
Not pythagorean theorem again... (Actually, I shouldn't be complaining because I didn't do the recent POTWs.) Anyways, the problem states that one person goes north and another goes east. This means that when they are 130 km away, the line must be diagonal, or else it would two numbers. This forms a triangle, since you start from the end of the distance traveled north, and go to the end of the distance traveled east. Drawing it out makes it a lot easier.
The diagonal line would be 130 km, so this creates a Pythagorean Theorem.
However... we don't have the two leg lengths yet, so this means that we have to solve for them. Since the formula for the Pythagorean theorem is a squared + b squared = c squared, this means that we should implement this into our problem. 130^2 = 16900. Now, we can square the two distances given for the legs, 24 km/h and 32 km/h. This helps with solving for the legs because soon we can compare the ratios. 24^2 = 6084 km and 32^2 = 10816 km. Using the ratio, we simplify to 9:16. Then, we can use 16900 divided by (9+16) to get 676, then multiply it by 9 and 16 to get the lengths. This gives me 6084 and 10816, but the value has not been square rooted yet, so I get 78 km and 104 km. However, it also asks what time it will be, so I will calculate using one of the people. For example, Gail goes for 78 km, but to figure out how much time it takes I will figure out how many km she bikes for in a minute. 24/60 = 0.4. This means that we can figure out how long it takes for Gail to bike 78 km going 0.4 km a minute. THe answer would be 78/0.4 = 195, simplifying to 3 hours and 15 minutes, making the answer 4:30. -Alan
Simple misclick on a comment lost most of my sentences I wrote down so I'm going to be brief. Dusty travels East, Gail travels North I can use Pythagorean theorem to find the slant as the distance between the two (assuming there is no change in direction ever). a^2 + b^2 = c^2 Plug in 130 km a^2 + b^2 = 130^ a^2 + b^2 = 19600 Use k as a variable (24k)^2 + (32k)^2 = 19600 (576k)^2 + (1024k)^2 = 16900 (1600k)^2 = 16900 (16k)^2= 169 k^2= 169/16 The above can't be divided anymore, so just simplify it k = 13/4 13/4 = 3.25 Using 60 minute time instead of 100, 3.25 becomes 3 hours and 15 minutes Gail = 24 * 13/4 = 76 km Dusty = 32 * 13/4 = 104 km CHECK: 104 + 76 = 130 km 1:15 + 3h 15m = 4:30 ish Therefore they will be 130 km apart at around 4:30
Both Gail Storm and Dusty Storm start biking at 1:15 pm. I already know the distance and time so I need to find the speed.
It takes 3 hours and 15 minutes for both of them to bike to a point where they are exactly 130 km away from each other. Gail: 24*3.25= 78 km Dusty: 32*3.25= 104 km 1:15 + 3 hours 15 minutes = 4:30 PM Therefore, they will be 130 km away from each other at 4:30 PM.
Grade 7/8 POTW
ReplyDeleteFirstly, I will list some things we know.
- Time 1:15
- Same starting point
- Gail Storm North 24 km/h
- Dusty Storm East 32 km/h
- When will there be a distance of 130 km.
Firstly, if we draw lines to show how far they are from each other as well as the starting point, you form a right triangle with the hypotenuse being the distance away from each other. This is probably the right track (ha ha) since this is the geometry unit as well. Anyways a^2 + b^2 = c^2. a = Gail Storm traveling distance, b = Dusty Storm travelling distance, c = distance from each other. From here we just see how large c^2. Right now, c = 130 km. 130^2 = 16900. Now, instead of trial and error, I'll find the ratio of Dusty and Gail's speeds as this can find the answer immediately.
Gail to Dusty = 24:32 = 3:4. However, this is without squaring it which means we have to square each number. 24^2 = 576, 32^2 = 1024. 576:1024 = 9:16. 9:16 of 16900. This is one way to continue from here. 9 + 16 = 25. 16900/25 = 676. 676 x 9 = 6084. 676 x 16 = 10816. The square root of 6084 is 78. The square root of 10816 is 104. Now we divide. 78/24 = 3.25. 104/32 = 3.25. They are both the same which proves this is most likely correct. 104 x 0.75 (3/4) = 78. More proof it is correct as it suits the ratio from before. 3.25 hours = 3 hours and 15 minutes (.25 = 1/4 = 1/4 of 60 = 15). 1:15 + 3 hours and 15 minutes = 4:30.
They will have a distance of 130 km at 4:30.
Dusty Storm traveled 104 km while Gail Storm traveled 78 km.
POTW:
ReplyDeleteInfo:
- Gail- 24 km/h- north
- Dusty- 32 km/h- east
- What time will they be 130 km apart? And how far will they have travelled?
Since both of them are travelling north and east (assuming that they do not change directions or slant slightly), they are riding two of the lengths of the triangle. The slant would be the distance between the two bikers. Using the Pythagorean Theorom, we can reverse it to find the other two values.
a^2 + b^2 = c^2.
a^2 + b^2 = 130^2
a^2 + b^2 = 16900
Using x as the multiple:
(24t)^2 + (32t)^2 = 130^2
(576t)^2 + (1024)^2 = 16900
(1600t) ^2 = 16900
(16t)^2= 169
t^2= 169/16 (this cannot be divided)
Simplify
13/4=3.25 Since we are dealing with 60 minutes is a whole: 3 hours 15 min
t= 13/4
24 x 13/4 = 78 = Gail
32 x 13/4 = 104 = Dusty
1:!5 + 3 h 15 min = 4:30 pm.
(The dedication to bike for 3 hours plus...)
Not pythagorean theorem again...
ReplyDelete(Actually, I shouldn't be complaining because I didn't do the recent POTWs.)
Anyways, the problem states that one person goes north and another goes east. This means that when they are 130 km away, the line must be diagonal, or else it would two numbers. This forms a triangle, since you start from the end of the distance traveled north, and go to the end of the distance traveled east. Drawing it out makes it a lot easier.
The diagonal line would be 130 km, so this creates a Pythagorean Theorem.
However... we don't have the two leg lengths yet, so this means that we have to solve for them.
Since the formula for the Pythagorean theorem is a squared + b squared = c squared, this means that we should implement this into our problem.
130^2 = 16900.
Now, we can square the two distances given for the legs, 24 km/h and 32 km/h. This helps with solving for the legs because soon we can compare the ratios.
24^2 = 6084 km and 32^2 = 10816 km. Using the ratio, we simplify to 9:16.
Then, we can use 16900 divided by (9+16) to get 676, then multiply it by 9 and 16 to get the lengths. This gives me 6084 and 10816, but the value has not been square rooted yet, so I get 78 km and 104 km. However, it also asks what time it will be, so I will calculate using one of the people.
For example, Gail goes for 78 km, but to figure out how much time it takes I will figure out how many km she bikes for in a minute. 24/60 = 0.4. This means that we can figure out how long it takes for Gail to bike 78 km going 0.4 km a minute. THe answer would be 78/0.4 = 195, simplifying to 3 hours and 15 minutes, making the answer 4:30.
-Alan
Grade 8 POTW
ReplyDeleteGail and Dusty will be 130 kilometres apart at around 4:30 pm, having biked for 3 hours and 15 minutes each. I did my work on paper.
You didn't say how far each of them had ridden.
DeleteThey will have be130 km apart at 4:30 p.m.
ReplyDeleteDusty Storm traveled 104 km while Gail Storm traveled 78 km.
Simple misclick on a comment lost most of my sentences I wrote down so I'm going to be brief.
ReplyDeleteDusty travels East, Gail travels North
I can use Pythagorean theorem to find the slant as the distance between the two (assuming there is no change in direction ever).
a^2 + b^2 = c^2
Plug in 130 km
a^2 + b^2 = 130^
a^2 + b^2 = 19600
Use k as a variable
(24k)^2 + (32k)^2 = 19600
(576k)^2 + (1024k)^2 = 16900
(1600k)^2 = 16900
(16k)^2= 169
k^2= 169/16
The above can't be divided anymore, so just simplify it
k = 13/4
13/4 = 3.25
Using 60 minute time instead of 100, 3.25 becomes 3 hours and 15 minutes
Gail = 24 * 13/4 = 76 km
Dusty = 32 * 13/4 = 104 km
CHECK: 104 + 76 = 130 km
1:15 + 3h 15m = 4:30 ish
Therefore they will be 130 km apart at around 4:30
They are 130 km apart at approximately 16:30.
ReplyDeleteBoth Gail Storm and Dusty Storm start biking at 1:15 pm.
ReplyDeleteI already know the distance and time so I need to find the speed.
It takes 3 hours and 15 minutes for both of them to bike to a point where they are exactly 130 km away from each other.
Gail: 24*3.25= 78 km
Dusty: 32*3.25= 104 km
1:15 + 3 hours 15 minutes = 4:30 PM
Therefore, they will be 130 km away from each other at 4:30 PM.