All Answers 355925 Answers

Polynomial Interpolation

$q(x) = -\frac{x^2}{10} + \frac{17}{10} x - 3$

$q(15) = -\frac{15^2}{10} + \frac{17}{10} \cdot 15- 3 = 0$

Calvin and Susie had some game cards at first. Calvin gave 1/4 of his cards to Susie. After that, Susie gave 1/5 of her cards to Calvin. In the end, Calvin had 45 cards and Susie had 36 cards. How many cards did Calvin have at first?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Calvin's cards => x     Susie's cards =>  y            (At first)

x + y = 81

x = 81 - y             

1/4(81 - y)        Susie received from Calvin

1/5[y + 1/4(81 - y)]           Susie gave to Calvin

4/5[y + 1/4(81 - y)] = 36          y = 33

x = 81 - 33

x = 48        Cards that Calvin had at first.

https://web2.0calc.com/questions/3-questions-on-algebra_1

3.5 sq units

Split the quadrilateral into two triangles using the y-axis as a line. You now have two triangles one with 2x3x1/2 = 3 sq units and 1 with 1 x 1 x1/2 = 0.5 units. Adding these up gives you your area of 3.5 square units

Formula for the volume of a sphere: 4/3(pi)(r)³

= 4/3(pi)2³ + 4/3(pi)3³ + 4/3(pi)6³ = 10 2/3(pi) + 36(pi) + 288(pi) = 334 2/3(pi)

y = (slope)x + (y-intercept)

Slope = y₂ - y₁ / x₂ - x₁

          = 11- (-17) / -5 - (-2)

          = 28/-3 or -9.3333333 or -9 1/3

Y-intercept = 2(-9 1/3) + -17 = -35 2/3

So...

The equation is y = -28/3x - 35 2/3

Where is the pic?

This question has been done to death!

Let the pieces be   x,  y-x  and  5-y       (that adds up to 5)

Now draw a contour probability map.

5 5/8 = 45 / 8      45/8 * 1/2 = 45/16 = 2  13/16

Original volume = l w h

  new volume = 1.1 l    1.1 w   1.1 h

                       1.331  l w h        a 33% increase

$\ell^2 = 180$

Let the number of minutes before noon ==M

Convert hours to minutes as follows:

12 x 60 ==720 minutes

9  x  60 ==540 minutes

720 - 40 - M ==540 + 3M

680 - M ==540 + 3M

680 - 540 ==3M + M

140 == 4M

M ==140 / 4 == 35 minutes to 12:00 noon

Check: 35 minutes to 12:00 noon is: 11:25 AM

40 minutes ago from 11:25 AM ==11:25  -  40 minutes ==10:45 AM

10:45 AM  ==60 + 45 ==105 minutes after 9:00 AM

3  x  35 minutes ==105 minutes after 9:00 AM - which balances

4 dollars

Is he/she selling something discounted?

Ron paid $14 for a book and 4 pens. The price of each pen was 5/8 the price of the book. How much did Ron pay for the book?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Price of 4 pens => x

Price of a book => 14-x

14 - x = 14 - 4[5/8 (14 - x)]

x = 10

Ron paid $4 for a book 

Max  Perimeter = 100 + 2(25√8)

Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is four inches. What is the original radius in inches?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

V = r²*pi*h

(6+r)² *pi * 4 = r² * pi * 10

The original radius=>    r = 2(2 + √10)

In which quadrant(s) do the graphs of y=x^2  and y=3x+10 intersect?

Hello Guest!

\(y=x^2=3x+10\\ x^2=3x+10\\ x^2-3x-10=0\\ x=1.5\pm \sqrt{2.25+10}\\ x\in \{-2,5\}\\ y\ \in\ \{\ 4,25\}\)

They intersect in quadrants I and II.

  !

It is 3/4 * the circumference of the circle  + 2 radii.

The formula for the circumference of a circle is 2pi*r

Now you can do it.

Find the equation of the line that passes through the points \(P_2\)(-2,1) and \(P_1\)(-3,3).

Hello Guest!

\(m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{1-3}{-2-(-3)}\)

\(m=-2\)

\(y=m(x-x_1)+y_1=-2(x-(-3))+3=-2x-6+3\)

\(y=-2x-3\)

  !

 y=16/(5+3x)

inverse

x=16/(5-3y)

-2= 16/(5-3y)

-2(5-3y)=16

3y-5 = -8

3y = -3

y=-1

\(f^{-1}(-2)=-1\)

A quadratic function with zeros at  a  and  b  is:

y  =  (x - a)(x - b)

Then we can multiply out the right side of that equation, or "FOIL" it, to get:

y  =  x² - bx - ax + ab

And then continue to simplify the right side until it looks like a nice quadratic equation.

Using this "template", we can find a solution to each of these problems.

a)    y   =   (x - 3)(x - 5)   =   x² - 5x - 3x + 15   =   x² - 8x + 15

b)    y   =   (x + 4)(x - 3)   =   x² - 3x + 4x - 12   =   x² + x - 12

c)    y   =   \((x-\frac12)(x-\frac23)\ =\ x^2-\frac23x-\frac12x+\frac13\ =\ x^2-\frac76x+\frac13\)

d)    In this case, the one root is  2 + √3  and the other is  2 - √3 , so the quadratic equation is:

        y  =  ( x - (2 + √3) )( x - (2 - √3) )

        y  =  ( x - 2 - √3 )( x - 2 + √3 )

        y  =  x² - 4x + 1

Can you figure out the last one? It might be the trickiest one, so if you need more help on it just ask!

Find the least positive four-digit solution to the following system of congruences.

7x = 21 (mod 14)

2x + 18 = 16 (mod 9)

7x=21 mod14

x=3 mod14

x=14p+3

2x+18=16 mod9

2x=-2  mod9

x=-1    mod9

x=9q-1

9q-1 = 14p+3

9q - 14p = 4

-------------------------

9=-1*(-14) - 5

-14=  3(-5)+1

-14 - 3(-5) = 1

-14 - 3 [9+1*(-14)] = 1

(-14) - 3*9 -3(-14) = 1

-2(-14) -3(9) = 1

9(-3) -14(-2)=1

9(-12) -14(-8)=4

9(-12+14g) - 14(-8+9g) = 4

so

q =-12+14g             and    p =-8+9g

x = 9q-1                  and     x= 14p+3

x= 9(-12+14g)-1     and     x= 14(-8+9g)+3

x= -109 +9*14g      and     x = -109 + 14*9g      they are the same so it is right.

x= -109 +126g

I want the smallest 4 digit value of x

-109+126g>1000

126g > 1109

g > 8.8

g=9

x = -109+126*9

x= 1025

Check your numbers. 

sketch them using Desmos