Solve for B
6=10B+12C

The problem is an optimization problem in mathematics where one must use algebra and calculus to maximize the area of a rectangular parking lot with a given amount of fencing along three sides. By setting up a function for the area in terms of width and differentiating, the maximum area can be found.

The problem presented involves finding the maximum area of a rectangular parking lot given a certain amount of fencing, when one side of the rectangle requires no fencing. This is a classic optimization problem in mathematics which involves the use of algebra and calculus. By setting up a function for the area of the rectangle in terms of one variable and differentiating to find the maximum value, we can determine the solution.

Lets assume that the length of the lot that is along the busy street doesn't require fencing, and let the width of the lot be w and the length be l. Since fencing is only required for three sides, and we have 2,200 feet of fencing, the perimeter equation is 2w + l = 2,200. Solving for l, we get l = 2,200 - 2w. The area A of the parking lot will be given by the equation A = w * l. Substituting the expression for l into the area equation, we get A = w(2,200 - 2w) or A = 2,200w - 2w^2. To find the maximum area, we take the derivative of A with respect to w, set it to zero, and solve for w. Once the width is found, we can find the length using the perimeter equation and thus calculate the maximum area of the lot.

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To calculate S(4), substitute the value of t with 4 into the function S(t), which results in S(4) = 9(4) - 4, giving the final answer of 32.

To find S(4), we need to substitute the value of t with 4 in the function S(t) = 9t - 4. This is a simple evaluation of a function for a given input.

Step-by-step calculation:

Therefore, S(4) is 32.

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The function f(x) varies directly with x and f(x) is 250 when x is 50, so

f(x) is 25 when x=5

the function f(x) varies directly with x and

f(x)=250 when x=50

when x=5  f(x) = 5*5 = 25

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Answer: 0.65a = 39

Step-by-step explanation:

To find the height of the triangle with an area of 64 square kilometers and a base twice its height, we use the area formula: A = (1/2) × base × height, which leads us to determine that the height is 8 kilometers.

The question given involves finding the height of a triangle when the area is known and the relation between the base and height is given. Since the base is twice the height (b = 2h), and we know the formula for the area of a triangle is A = (1/2) × base × height, we can substitute the known area and the base-height relationship into this formula to solve for the height.

The area (A) is given as 64 square kilometers, which leads to the equation:

A = (1/2) × b × h = (1/2) × (2h) × h = h²,

Therefore, with A = 64 km², we obtain:

64 = h².

Taking the square root of both sides gives us the height (h):

h = √64 km = 8 km.

So the height of the triangle is 8 kilometers.

The crescent wrench is a type of lever, a simple machine that amplifies force to provide greater torque on fasteners with less input effort.

The crescent wrench is a type of simple machine, specifically it is a form of a lever. When you use a crescent wrench, you apply force to the handle, which then applies force to the object you are turning. The pivot point, or fulcrum, is the adjustable jaw which grips the nut or bolt. This allows the user to exert a greater torque on the fastener with less force than would be needed if just using one's hands.

Like other levers, the crescent wrench allows for a mechanical advantage, which is the ratio of output to input forces for any simple machine. By lengthening the handle of the wrench (the effort arm), we achieve a greater mechanical advantage, and thus, can apply more torque to the bolt or nut we are trying to turn.

The time taken to assemble 125 amplifiers is required.

Time taken to assemble 125 amplifiers is 750 minutes or 12 hours and 30 minutes.

Time taken to assemble 5 amplifiers is 30 minutes

[tex]5\ \text{amplifiers}=30\ \text{minutes}[/tex]

So,

[tex]1\ \text{amplifier}=\dfrac{30}{5}=6\ \text{minutes}[/tex]

[tex]125\ \text{amplifiers}=125\times 6=750\ \text{minutes}=\dfrac{750}{60}=12.5\ \text{hours}[/tex]

So, time taken to assemble 125 amplifiers is 750 minutes or 12 hours and 30 minutes.

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The volume of the solid obtained by rotating the region bounded by the curves about the x-axis is found using cylindrical shells, resulting in a total volume of 36π cubic units.

To find the volume of the solid obtained by rotating the region bounded by the curves xy = 9, x = 0, y = 9, and y = 11 about the x-axis, we will utilize the method of cylindrical shells. The curves define a region in the xy-plane which, when rotated about the x-axis, forms a solid with varying radii. To apply the method, we first need to express the variable x in terms of y from xy = 9 to obtain x = 9/y.

Next, the volume of a representative cylindrical shell with thickness dy and height x = 9/y is given by the circumference of the shell (2πy) times the height (9/y) times the thickness (dy). This leads to the volume element dV = 2πy(9/y)dy = 18πdy. Integrating this volume element with respect to y between the limits 9 and 11 yields the total volume.

Using the integral calculus, the volume V is:
V = ∫911 18π dy = 18π [y]∫911 = 18π(11 - 9) = 36π cubic units.

Thus, the volume of the cylinder is 36π cubic units.

Answer:

  b)  0.25

Step-by-step explanation:

The required probability is ...

  (24 -20)/(36 -20) = 4/16 = 1/4 = 0.25

_____

Comment on the attached graph

The red line is the probability distribution function (pdf) for this uniform distribution. The green line is the cumulative distribution function (cdf) for this uniform distribution. It is the integral of the pdf.

The probability of interest is the value of the cdf at x=24. That value is 0.25, meaning the area under the pdf curve to the left of x=24 is 1/4 of the total area under the pdf curve. That proportion is what is calculated above.

Jay's error was in the order of operations. The correct result is 17.

Jay's error lies in how he interpreted the expression. According to the order of operations (PEMDAS/BODMAS), multiplication and division should be performed before addition and subtraction.

So, the correct first step is to perform the division [tex]\( \frac{12}{3} = 4 \)[/tex]and then add 3 to the result, not add 3 and 12 together first.

The correct first step:

[tex]\[ 3 \times (3 + \frac{12}{3}) - 4 \][/tex]

[tex]\[ = 3 \times (3 + 4) - 4 \][/tex]

[tex]\[ = 3 \times 7 - 4 \][/tex]

= 21 - 4

= 17

Therefore, Jay's error was in the order of operations. The correct result is 17.

Answer:

(a) 10π cm

(b) r = 5 cm

(c) h = √11 cm

Step-by-step explanation:

(a) The surface of the opening the cup is circular in shape and a sector is cut from circle is calculate by formula,

C = r × θ

here, r = 6 cm

and θ = [tex]\frac{5\pi}{3}[/tex]

⇒ [tex]C = 6 \times \frac{5\pi}{3} = 10\pi[/tex]

(b) For finding the radius of the cup,

As the circumference of the circle is 10π

and we know that area of cone is calculate by formula, 2πr

⇒ 2πr = 10π

⇒ r = 5 cm

(c) In cone we know slant height(l) = 6 cm

Radius = 5 cm

Thus, using Pythagoras theorem,

l² = h² + r²

⇒ h² = l² - r²

⇒ h² = 36 - 25

⇒ h = √11 cm