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 #1
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One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?  

 

 

To find b, consider                                                  ab4  =  48  

 

We will divide both sides by ab.  

 

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab4         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab4 = (ab) * (b3)  

 

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b3           2  

                                                                             ——   =   ——  

                                                                                1            3  

 

 

                                                                                 b   =   cube root of (2 / 3)  

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15 jun 2023
 #1
avatar+752 
0

 

Will and Grace are canoeing on a lake.  Will rows at 50 meters per minute and Grace rows at 20 meters per minute. Will starts rowing at 2 p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at 2 p.m. If they always row directly towards each other, and the lake is 3800 meters across from the west side of the lake to the east side, at what time will the two meet?  

 

 

Will's rate is 50 m/min.  

Grace's rate is 20 m/min.  

 

They're rowing toward each other,  

so their rate of closure is 70 m/min.  

 

The lake is 3800 meters across.  

 

The time it will take them to meet is given by  

                                                                                    Distance  

                                                                            T  =  –––––––  

                                                                                       Rate  

 

                                                                                    3800 m  

                                                                            T  =  –––––––  

                                                                                    70 m/min  

 

The time it takes them to meet                             T  =  54.29 min  

 

The time on the clock is          2:00 pm + 54.29 min  =  2:54:17 pm 

 

(The :17 is because 0.29 minute is 17 seconds.)  

14 jun 2023
 #2
avatar+752 
0

 

Hello Monica.  Good work.  Excellent, even :-)   When I answered this same problem some time back, I took a slightly different path, but we both ended up at the same place, namely 104 gallons of paint.  I was happy to see someone else's answer was consistent with mine, because there was a "solution" even earlier than mine that ended with a different (read incorrect) answer.  Here's how I did it:    

 

 

You're going to have to convert those meters to square feet,  

so I think it's better to do it from the start.  Even though we'll  

have fractions to contend with, a calculator makes that easy.  

 

                                                                       12 m  

1 foot is 0.3048 meter, so the height is       –––––––––  =  39.3701 ft   

                                                                   0.3048 m/ft  

 

                                                                       120 m  

1 foot is 0.3048 meter, so the length is       –––––––––  =  393.7007 ft   

                                                                   0.3048 m/ft  

 

Area is height times length     (39.3701 ft)(393.7007 ft)  =  15,500.0359 sq ft  

 

1 gallon will cover 150 sq ft  

so divide that into the area               15,500.0359 sq ft  

                                                        –––––––––––––––  =  103.3335 gallons  

                                                           150 sq ft/gallon  

 

You can't buy just part of a gallon of paint,  

so we have to round that up to whole gallons                       104 gallons  

.

14 jun 2023
 #2
avatar+752 
+1

 

In regard to #1 above "...the value of k that makes the equation have exactly one solution is k = 3/2."  

 

I think that solution was for the value of x, instead of for the value of k.  

 

I followed as far as "...one solution when the discriminant b^2 - 4ac = 0."  So far, so good. 
 

At that point, I think I would substitute values in             b2 – 4ac  =  0  

 

                                                                            (6)2 – (4)(2)(k)  =  0  

 

                                                                                        36 – 8k  =  0  

 

                                                                                                8k  =  36  

 

                                                                                                  k  =  36/8  

 

                                                                                                  k  =  9/2  

I don't know how to check the veracity  

of this answer.  If I'm wrong, I would truly  

appreciate someone showing me where.  

.

12 jun 2023