Let us find the edge length
Draw a segment from the apex of the tetrahedron perpendicular to the base.....this is the height....call the point where this segment intersects the base, A
Draw a segment from one of the bottom vertexes (call this point B) to A
Find the midpoint of one of the sides connecting to B....label this point C
And BC = (1/2) the side length = (1/2)s
So.....we have the right triangle ABC lying on the base
And we have this relationship
cos 30° = ( BC) / (BA)
cos (30°) = (1/2)s / (BA)
BA = (1/2)s / [√ 3 / 2] = s / √ 3
So......we have another right triangle...one leg is the height, one leg is BA and the hypotenuse is the side length......and we have......
√ [ 20^2 + BA^2] = s
√ [ 20^2 + (s/√3)^2 ] = s
√ [(1200 + s^2) / 3 ] = s
[1200 + s^2] / 3 = s^2
1200 + s^2 = 3s^2
1200 = 2s^2
600 = s^2
√ 600 = s
10√ 6 = s ≈ 24.49 inches
We can solve this with a graph [ linear programming ]
Let x = number of $75 tires to be sold and y = the number of $ 85 tires to be sold
Here are the constraints to be graphed :
x + y ≤ 300
3y ≤ 2x
And the objective function to be maximized is this
75x + 85y
A look at the graph here : https://www.desmos.com/calculator/5dtkxwrxak will show that the max occurs at the corner points of the intersection of the two inequalities
There is only one corner point at ( x , y) = (180, 120)
Putting this into the objective function produces
75(180) + 85 (120) = $ 23700
log √[5x] + log √[20x]
Note that we can use the properties
log a + log b = log [ a * b ] and log ab = b*log a
So we can write
log √[5x] + log √[20x] =
log (5x)1/2 + log (20x)1/2 =
(1/2) [ log (5x) + log (20x) ] =
(1/2) [ log (5x * 20x) ] =
(1/2) log ( 100 x2) =
log (100x2)1/2 =
log √ [ 100x2 ]
log (10x) which can also be simplified as
log 10 + log x =
1 + log x
The sum of two consecutive integers is no more than 209.
What is the larger of the two integers?
N + N +1 ≤ 209
2N + 1 ≤ 209 subtract 1 from both sides
2N ≤ 208 divide both sides by 2
N ≤ 104
The larger of the two integers = N + 1 = 105
A figure skater needs a total score of at least 90 to move on to the next round. The total score is the average of four judges’ scores. The first three judges’ scores were 83, 88, and 92. The figure skater made it to the next round.
What is the minimum score the fourth judge could have given?
We have that
[ 83 + 88 + 92 + M ] / 4 ≥ 90 multiply both sides by 4
83 + 88 + 92 + M ≥ 360 simplify
263 + M ≥ 360 subtract 263 from both sides
M ≥ 97 → the skater needs to get a score of 97 [or more ]
The length of a rectangle is five times its width. The perimeter of the rectangle is at most 96 cm.
Which inequality models the relationship between the width and the perimeter of the rectangle?
L = 5W
Perimeter = 2 [ W + L ]
So......substituting for the width, we have
2 [ W + 5W ] ≤ 96 simplify
2 [ 6W ] ≤ 96
12W ≤ 96
Reiko started a business selling home medical supplies. She spent $6500 to obtain her merchandise, and it costs her $550 per week for general expenses. She earns $900 per week in sales.
What is the minimum number of weeks it will take for Reiko to make a profit?
We actually just need to equate costs and sales....so we have....
6500 + 550W = 900W where W is the number of weeks that we are trying to find
Subtract 550W from both sides
6500 = 350W
Divide both sides by 350
18.57 ≈ W .....so...it will take her 19 weeks to start making a profit
Ron is five years older than twice his cousin Pat’s age. The sum of their ages is less than 35.
What is the greatest age that Pat could be?
Let Pat's age = P
Ron's age = 2P + 5
And we know that
P + (2P + 5) < 35 simplify
3P + 5 < 35 subtract 5 from both sides
3P < 30 divide both sides by 3
P < 10 → Pat's greatest possible age is 9