Answer:
1) 15cm
2) left projection/h = h/right projection
Step-by-step explanation:
Question:
1) The projections of the legs of a right triangle on the hypotenuse have lengths of 9 cm and 25 cm. Find the length of the height at the top of the right angle.
2) In a right triangle the length of the hypotenuse is 34 cm, and the lengths of the projections of the legs on the hypotenuse are directly proportional to the numbers 0, (6) and 0.75. Calculate the length of the height corresponding to the hypotenuse.
Solution
1) The length of the height of a right angle triangle is also called the altitude.
Since there are no diagrams in the question, I sent a diagram of the right angle as an attachment to the solution.
The projections of the legs are 25cm and 9cm.
Hence, the longer projection length AD = 25cm and the shorter projection length DB = 9cm
In a right triangle, the altitude (height) drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the
geometric mean of these two segments (the two projections) and it's given by:
left projection/h = h/right projection
AD/h = h/DB
25/h = h/9
Cross multiply
h^2 = 25×9
h =√225 = 15cm
The length of the height at the top of the triangle = 15cm
2) Length of hypotenuse = 34
From the question, the lengths of the projections of the legs on the hypotenuse are directly proportional to the numbers 0, (6) and 0.75.
There is an error with the figures because the sum of the length of the projection of the legs should be equal to the hypotenuse but it isn't in this case.
To calculate the length of the height corresponding to the hypotenuse, we would use the same formula above.
left projection/h = h/right projection
To find each leg using question 1 above, each leg of the triangle is the mean proportional between the hypotenuse and the part of the hypotenuse directly below the leg.
Hypotenuse =34cm
Hyp/leg = leg/part
To find leg y, part for leg y = 25cm
34/y = y/25
y^2 = 34×25 = 850
y = √850 = 29.2cm
To find leg x, part for leg x = 9cm
34/y = y/9
y^2 = 34×9 = 306
y = √306 = 17.5cm
1) The outcomes for rolling two dice, the sample space, is as follows:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
There are 36 outcomes in the sample space.
2) The ways to roll an odd sum when rolling two dice are:
(1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (3, 6), (4, 1), (4, 3), (4, 5), (5, 2), (5, 4), (5, 6), (6, 1), (6, 3), (6, 5). There are 18 outcomes in this event.
3) The probability of rolling an odd sum is 18/36 = 1/2 = 0.5
Answer:
The value of x is 
hours.
Step-by-step explanation:
Machine A = 5 hours
Machine B = x hours
Machine A and B = 2 hours
Using the formula: 
where:
T is the time spend by both machine
A is the time spend by machine A
B is the time spend by machine B
Let multiply the entire problem by the common denominator (5B)
2x + 10 = 5x
Collect the like terms
10 = 5x - 2x
10 = 3x
3x = 10
Divide both side by the coefficient of x (3)
hours.
Therefore, Machine B will fill the same lot in hours.
You do the implcit differentation, then solve for y' and check where this is defined.
In your case: Differentiate implicitly: 2xy + x²y' - y² - x*2yy' = 0
Solve for y': y'(x²-2xy) +2xy - y² = 0
y' = (2xy-y²) / (x²-2xy)
Check where defined: y' is not defined if the denominator becomes zero, i.e.
x² - 2xy = 0 x(x - 2y) = 0
This has formal solutions x=0 and y=x/2. Now we check whether these values are possible for the initially given definition of y:
0^2*y - 0*y^2 =? 4 0 =? 4
This is impossible, hence the function is not defined for 0, and we can disregard this.
x^2*(x/2) - x(x/2)^2 =? 4 x^3/2 - x^3/4 = 4 x^3/4 = 4 x^3=16 x^3 = 16 x = cubicroot(16)
This is a possible value for y, so we have a point where y is defined, but not y'.
The solution to all of it is hence D - { cubicroot(16) }, where D is the domain of y (which nobody has asked for in this example :-).
(Actually, the check whether 0 is in D is superfluous: If you write as solution D - { 0, cubicroot(16) }, this is also correct - only it so happens that 0 is not in D, so the set difference cannot take it out of there ...).
If someone asks for that D, you have to solve the definition for y and find that domain - I don't know of any [general] way to find the domain without solving for the explicit function).
