Questions   
Sort: 
 #2
avatar+131 
0
Jun 28, 2023
 #1
avatar+2435 
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  

 

Note that (0.29 min • 60 sec/min)  =  17 seconds 

 

The time on the clock is                                             2:54:17 pm 

.

Jun 28, 2023
 #1
avatar+2435 
0

 

I have to paint one side of a wall.  The wall is 12 meters tall and 120 meters long.  Each gallon of paint covers 150 square feet. If a foot is approximately 0.3048 meters, then what is the smallest whole number of gallons I can buy and have enough paint to cover the whole wall?  

 

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  

.

Jun 28, 2023
 #1
avatar+2435 
0

 

I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?   

 

You mean 45 minutes earlier.  Obviously, if you drive faster, you get there faster.  

 

This problem makes use of

the following relationship:                   Distance = Velocity x Time 

 

                                                           D  =  V • T  

 

case 1                                                 D  =  (40) • (T)  

 

case 2                                                 D  =  (50) • (T – 45)  

 

Since the Distance, D, is the  

same for both cases, let's set       

the "V•T"s equal to each other.             (50)(T – 45)  =  (40)(T)  

 

                                                               50T – 2250  =  40T  

 

Subtract 40T from both sides                  10T – 2250  =  0  

 

Add 2250 to both sides                                       10T  =  2250  

 

Divide both sides by 10                                           T  =  225   (this is in minutes)  

 

Divide minutes by 60 to get hours          225 minutes  =  3.75 hours  

 

Plug this T back into original equation                     D  =  (40 mi/hr) • (3.75 hr)  =  150 miles  

.

Jun 28, 2023
 #1
avatar+4 
0

Hello,

 

To find the distance between the foci of the ellipse, we can use the properties of ellipses and the given information.  TargetPayandBenefits

First, let's denote the coordinates of the center of the ellipse as (h, k), where h represents the horizontal shift and k represents the vertical shift. Since the ellipse is tangent to the x-axis at (a, 0), we know that the distance between the center and the x-axis is a, which means k = a.

Since the ellipse is also tangent to the y-axis at (0, b), we know that the distance between the center and the y-axis is b, which means h = b.

Now, we have the center coordinates as (a, a). The distance between the foci of an ellipse can be calculated using the formula c = √(a^2 - b^2), where c represents the distance between the center and each focus.

Substituting h = b = a into the formula, we get c = √(a^2 - a^2) = √0 = 0.

Therefore, the distance between the foci of the given ellipse is 0.

Jun 28, 2023
 #1
avatar
0

The number of ways to place 8 counters in the grid so that each row contains exactly two counters is the same as the number of ways to place 8 counters in the grid so that each column contains exactly two counters. This is because we can simply rotate the grid 90 degrees to get from one case to the other.

So, let's consider the problem of placing 8 counters in the grid so that each column contains exactly two counters. We can think of this as a placement problem, where we have 8 counters and 4 columns, and we want to place the counters in the columns so that no column has more than 2 counters.

There are a few ways to solve this problem. One way is to use casework. We can consider the following cases:

Case 1: 2 counters in one column and 1 counter in each of the other 3 columns.

Case 2: 1 counter in one column and 2 counters in each of the other 3 columns.

Case 3: 1 counter in each of 2 columns and 2 counters in each of the other 2 columns.

For each case, we can count the number of ways to place the counters. For example, in Case 1, there are 4 ways to choose which column will have 2 counters, and then there are 3 ways to choose which of the remaining 3 columns will have the first counter, and then 2 ways to choose which of the remaining 2 columns will have the second counter. So, there are 4×3×2=24 ways to place the counters in Case 1.

Similarly, there are 4×3=12 ways to place the counters in Case 2, and 4×2=8 ways to place the counters in Case 3.

So, the total number of ways to place the counters is 24+12+8=44​.

Jun 28, 2023

0 Online Users