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PolarNik [594]
3 years ago
11

The city is planning to add a fishpond to a neighborhood park the figure below is a scale drawing of the fishpond it's scale is

1/2 inch equals 15 feet
Mathematics
1 answer:
notsponge [240]3 years ago
5 0
I saw the figure of the fishpond. It composed of a rectangle and a circle. The circle is cut into two and each half is attached to the width of the rectangle making an oblong shaped fishpond.

Length of the rectangle: 2.5 inch
Width of the rectangle and diameter of the circle: 1 inch

1/2 inch equals 15 feet.

2.5 inches = 75 feet
1 inch = 30 feet

Area of a rectangle = 75 ft * 30 ft = 2,250 ft²
Area of a circle = 3.14 * (15ft)² = 3.14 * 225ft² = 706.50 ft²

Total Area = 2,250 ft² + 706.50 ft² = 2,956.50 ft²
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Find the exact length of the curve. 36y2 = (x2 − 4)3, 5 ≤ x ≤ 9, y ≥ 0
IrinaK [193]
We are looking for the length of a curve, also known as the arc length. Before we get to the formula for arc length, it would help if we re-wrote the equation in y = form.

We are given: 36 y^{2} =( x^{2} -4)^3
We divide by 36 and take the root of both sides to obtain: y = \sqrt{ \frac{( x^{2} -4)^3}{36} }

Note that the square root can be written as an exponent of 1/2 and so we can further simplify the above to obtain: y =  \frac{( x^{2} -4)^{3/2}}{6} }=( \frac{1}{6} )(x^{2} -4)^{3/2}}

Let's leave that for the moment and look at the formula for arc length. The formula is L= \int\limits^c_d {ds} where ds is defined differently for equations in rectangular form (which is what we have), polar form or parametric form.

Rectangular form is an equation using x and y where one variable is defined in terms of the other. We have y in terms of x. For this, we define ds as follows: ds= \sqrt{1+( \frac{dy}{dx})^2 } dx

As a note for a function x in terms of y simply switch each dx in the above to dy and vice versa.

As you can see from the formula we need to find dy/dx and square it. Let's do that now.

We can use the chain rule: bring down the 3/2, keep the parenthesis, raise it to the 3/2 - 1 and then take the derivative of what's inside (here x^2-4). More formally, we can let u=x^{2} -4 and then consider the derivative of u^{3/2}du. Either way, we obtain,

\frac{dy}{dx}=( \frac{1}{6})( x^{2} -4)^{1/2}(2x)=( \frac{x}{2})( x^{2} -4)^{1/2}

Looking at the formula for ds you see that dy/dx is squared so let's square the dy/dx we just found.
( \frac{dy}{dx}^2)=( \frac{x^2}{4})( x^{2} -4)= \frac{x^4-4 x^{2} }{4}

This means that in our case:
ds= \sqrt{1+\frac{x^4-4 x^{2} }{4}} dx
ds= \sqrt{\frac{4}{4}+\frac{x^4-4 x^{2} }{4}} dx
ds= \sqrt{\frac{x^4-4 x^{2}+4 }{4}} dx
ds= \sqrt{\frac{( x^{2} -2)^2 }{4}} dx
ds=  \frac{x^2-2}{2}dx =( \frac{1}{2} x^{2} -1)dx

Recall, the formula for arc length: L= \int\limits^c_d {ds}
Here, the limits of integration are given by 5 and 9 from the initial problem (the values of x over which we are computing the length of the curve). Putting it all together we have:

L= \int\limits^9_5 { \frac{1}{2} x^{2} -1 } \, dx = (\frac{1}{2}) ( \frac{x^3}{3}) -x evaluated from 9 to 5 (I cannot seem to get the notation here but usually it is a straight line with the 9 up top and the 5 on the bottom -- just like the integral with the 9 and 5 but a straight line instead). This means we plug 9 into the expression and from that subtract what we get when we plug 5 into the expression.

That is, [(\frac{1}{2}) ( \frac{9^3}{3}) -9]-([(\frac{1}{2}) ( \frac{5^3}{3}) -5]=( \frac{9^3}{6}-9)-( \frac{5^3}{6}-5})=\frac{290}{3}


8 0
3 years ago
An amusement park charges $25 to enter plus $2 per ride ticket. Which graph models the cost of a visit to this park?
KonstantinChe [14]

Answer:

Graph B

Step-by-step explanation:

6 0
3 years ago
What is degenerate circle
Viefleur [7K]
It is the circle which has its radius a 0units and it can also be called as a point circle.
4 0
3 years ago
Which of the following is a correct unit of weight: kilogram, newton, grams, slugs
AveGali [126]

Answer:

gram

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Use the net to find the surface area of the prism.
mel-nik [20]

Answer:

a. 96 in²

Step-by-step explanation:

The surface area of the prism = the sum of the area of each part of the net shown

✔️Area of the 2 equal triangles = 2(½*b*h)

b = 4 in

h = 3 in

Area of the two triangles = 2(½*4*3) = 12 in²

✔️Area of the rectangle 1 with the following dimensions:

L = 7 in

W = 3 in

Area = L*W = 7*3 = 21 in²

✔️Area of the rectangle 2 with the following dimensions:

L = 7 in

W = 4 in

Area = L*W = 7*4 = 28 in²

✔️Area of the rectangle 3 with the following dimensions:

L = 7 in

W = 5 in

Area = L*W = 7*5 = 35 in²

✅Surface area of the prism = 12 + 21 + 28 + 35 = 96 in²

4 0
2 years ago
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