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3 years ago

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The price of a skateboard is 5 dollars less than 3 times the price of a pair of knee pads. Let k represent the price of the knee

pads. Write an expression that gives the price of the skateboard

1 answer:

makkiz [27]3 years ago

8 0

Answer:

Price of skateboard= $3k-5$

Step-by-step explanation:

Given: The price of a skateboard is 5 dollars less than 3 times the price of a pair of knee pads.

Lets assume the price of the knee pads be "k".

As given, The price of a skateboard is 5 dollars less than 3 times the price of a pair of knee pads.

∴ y-intercept is 5 and slope will be 3.

using y intercept expression to form an expression.

we know, $y= mx+b$

∴ Price of skateboard= 3\times k-5

Price of skateboard= $3k-5$

Hence, expression for price of skateboard is $3k-5$

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A civil engineer is designing a public parking lot for the new town hall. The number of cars in each row should be six less than

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The new parking lot must hold twice as many cars as the previous parking lot. The previous parking lot could hold 56 cars. So this means the new parking lot must hold 2 x 56 = 112 cars

Let y represent the number of cars in each row, and x be the number of total rows in the parking lot. Since the number of cars in each row must be 6 less than the number of rows, we can write the equation as:

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Using the above two equations, the civil engineer can find the number of rows he should include in the new parking lot.

Using the value of y from equation 1 to 2, we get:

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3 years ago

A metal cylinder can with an open top and closed bottom is to have volume 4 cubic feet. Approximate the dimensions that require

Aleksandr-060686 [28]

Answer:

$r\approx 1.084\ feet$

$h\approx 1.084\ feet$

$\displaystyle A=11.07\ ft^2$

Step-by-step explanation:

<u>Optimizing With Derivatives </u>

The procedure to optimize a function (find its maximum or minimum) consists in :

Produce a function which depends on only one variable
Compute the first derivative and set it equal to 0
Find the values for the variable, called critical points
Compute the second derivative
Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum

We know a cylinder has a volume of 4 $ft^3$ . The volume of a cylinder is given by

$\displaystyle V=\pi r^2h$

Equating it to 4

$\displaystyle \pi r^2h=4$

Let's solve for h

$\displaystyle h=\frac{4}{\pi r^2}$

A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is

$\displaystyle A=\pi r^2+2\pi rh$

Replacing the formula of h

$\displaystyle A=\pi r^2+2\pi r \left (\frac{4}{\pi r^2}\right )$

Simplifying

$\displaystyle A=\pi r^2+\frac{8}{r}$

We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero

$\displaystyle A'=2\pi r-\frac{8}{r^2}=0$

Rearranging

$\displaystyle 2\pi r=\frac{8}{r^2}$

Solving for r

$\displaystyle r^3=\frac{4}{\pi }$

$\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet$

Computing h

$\displaystyle h=\frac{4}{\pi \ r^2}\approx 1.084\ feet$

We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative

$\displaystyle A''=2\pi+\frac{16}{r^3}$

We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.

The minimum area is

$\displaystyle A=\pi(1.084)^2+\frac{8}{1.084}$

$\boxed{ A=11.07\ ft^2}$

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3 years ago