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pishuonlain [190]
3 years ago
14

A ladder 6.5 m long rests against a vertical wall. If the foot of the ladder is 2.5m from the foot of the wall find upto how muc

h height does the ladder reach​
Mathematics
1 answer:
Burka [1]3 years ago
7 0

Answer:

The ladder reaches a height of 6 m.

Step-by-step explanation:

Given that,

Height of the ladder, H = 6.5 m

The foot of the ladder is 2.5 m from the foot of the wall.

We need to find the height does the ladder reach. If we consider a triangle in which hypotenuse is 6.5 m, foot of the ladder is 2.5 m then we need to find the height of the ladder i.e.

H^2=b^2+h^2\\\\h^2=H^2-b^2\\\\h^2=(6.5)^2-(2.5)^2\\\\h=6\ m

The ladder reaches a height of 6 m.

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Minimizing Packaging Costs If an open box has a square base and a volume of 107 in.3 and is constructed from a tin sheet, find t
uysha [10]

Answer:

The dimensions of the box that minimizes the amount of material of construction is

Square base = (5.98 × 5.98) in²

Height of the box = 2.99 in.

Step-by-step explanation:

Let the length, breadth and height of the box be x, z and y respectively.

Volume of the box = xyz = 107 in³

The box has a square base and an open top.

x = z

V = x²y = 107 in³

The task is to minimize the amount of material used in its construction, that is, minimize the surface area of the box.

Surface area of the box (open at the top) = xz + 2xy + 2yz

But x = z

S = x² + 2xy + 2xy = x² + 4xy

We're to minimize this function subject to the constraint that

x²y = 107

The constraint can be rewritten as

x²y - 107 = constraint

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

where λ = Lagrange factor, which can be a function of x and y

L(x,y,z) = x² + 4xy - λ(x²y - 107)

We then take the partial derivatives of the Lagrange function with respect to x, y and λ. Because these are turning points, at the turning points each of the partial derivatives is equal to 0.

(∂L/∂x) = 2x + 4y - 2λxy = 0

λ = (2x + 4y)/2xy = (1/y) + (2/x)

(∂L/∂y) = 4x - λx² = 0

λ = (4x)/x² = (4/x)

(∂L/∂λ) = x²y - 107 = 0

We can then equate the values of λ from the first 2 partial derivatives and solve for the values of x and y

(1/y) + (2/x) = (4/x)

(1/y) = (2/x)

x = 2y

Hence, at the point where the box has minimal area,

x = 2y

Putting these into the constraint equation or the solution of the third partial derivative,

x²y - 107 = 0

(2y)²y = 107

4y³ = 107

y³ = (107/4) = 26.75

y = ∛(26.75) = 2.99 in.

x = 2y = 2 × 2.99 = 5.98 in.

Hence, the dimensions of the box that minimizes the amount of material of construction is

Square base = (5.98 × 5.98) in²

Height of the box = 2.99 in.

Hope this Helps!!!

5 0
3 years ago
the cafeteria ordered 400 paper plates. they usef 226 at breakfast. they bought 100 more. Then, they used some plates for lunch.
vazorg [7]
196 Plates used for lunch. 400-226=174 174+100=274 274-78=196 Plates Used For Lunch
7 0
3 years ago
Read 2 more answers
What is the range of f(x) = 3/4x -4
shusha [124]

Answer:

The range is the resulting y-values we get after substituting all the possible x-values.

For the given function : f(x) = \frac{3}{4x} -4

<u>See the attached figure.</u>

The zeros of the denominator at x = 0

The domain is: (-∞,0)∪(0,∞)

<u>The range</u> of the function is the domain of the inverse function of f(x)

y = 3/(4x) - 4

y + 4 = 3/(4x)

4x = 3/(y+4)

x=\frac{3}{4(y+4)}

The zeros of the inverse function:

4(y+4) = 0

y + 4 = 0

y = -4

∴ The range is (-∞,-4)∪(-4,∞)

So, the answer is {y | y > –4} ∪ {y | y < – 4}

=========================================

<u>Note</u>: If the given function is : f(x) = \frac{3}{4} x-4

It will be first degree polynomial function.

Both of the domain and the range = all real numbers R

4 0
4 years ago
What is 1.04 as a fraction
Artemon [7]
My answer would be 95% correct but I give it to you anyways

The answer is.................... <span> 100 to give 104. TAAA DAHH</span>
5 0
3 years ago
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a man pays $225 each month into an account to pay his bill .how much does he pay into the account in three years?​
natta225 [31]

Answer:

$8,100

Step-by-step explanation:

He pays $225 for three years. There are 12 months in a year, so they are 36 months in 3 years. $225 a month times 36 months will get you $8,100.

7 0
3 years ago
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